基础：三个伟大的想法

Foundations: Three Great Ideas

只有三个想法——波、计算和像素——构成了数字光所有明显复杂性的基础。每个想法在直觉上都是简单、深刻和美丽的。这些是我们现代世界的技术基石，你不需要数学来理解它们。前三章（本书的第一部分）介绍了这些基本思想，以及使它们成为可能的人们的令人惊讶和引人入胜的故事。

Just three ideas—waves, computations, and pixels—underlie all the apparent complexity of Digital Light. Each idea is intuitively simple, profound, and beautiful. These are the technological cornerstones of our modern world, and you don’t need mathematics to understand them. The first three chapters (part I of this book) present these foundational ideas with the surprising and fascinating stories of the people who made them possible.

Waves 是一个模拟的想法。您可能知道音乐是由不同频率（音高）和幅度（响度）的同时声波组成的。两个世纪前，法国人约瑟夫·傅立叶将这一概念扩展到我们所有的感官体验。我们看到和听到的一切都是波的总和。都是音乐。在本书中，我将向您展示如何在视觉场景中观看音乐。

Waves are an analog idea. You probably know that music is made of simultaneous sound waves of different frequencies (pitches) and amplitudes (loudnesses). Two centuries ago the Frenchman Joseph Fourier extended that notion to all our sensory experience. Everything we see and hear is a sum of waves. It’s all music. In this book I show you how to see the music in a visual scene.

计算机是一种数字概念。使计算快速进行的机器是日常生活中数字化的典范。但是计算的概念只能追溯到 1936 年，当时英国人艾伦·图灵发明了它，以捕捉谨慎、精确过程的概念。这可能听起来乏味乏味，但后果绝不是这样。计算机是人类最具延展性的工具。还有他们惊人的速度是有史以来最重要的工程奇迹。以这样的速度，计算机将我们微不足道的人类可以做的事情放大到难以想象的程度。

Computers are a digital idea. Machines that make computations go fast are the very exemplars of the digital in ordinary life. But the idea of computation dates back only to 1936 when the Englishman Alan Turing invented it to capture the notion of a careful, precise process. That might sound plodding and boring, but the consequences are anything but. Computers are humankind’s most malleable tool. And their awesome speed is the most consequential engineering miracle of all time. With that speed, computers amplify what we puny humans can do by unimaginable amounts.

但是计算机的所有令人难以置信的、改变世界的能力实际上都归结为在两个状态之间小心翻转，通常被命名为 0 和 1。计算都是比特。这听起来可能微不足道，但我希望用计算中固有的意想不到的美丽和神秘来激发你的灵感。同样，不需要数学。

But all of the mind-boggling, world-changing power of computers really reduces to a careful flipping between two states, often named 0 and 1. Computation is all bits. That may sound trivial, but I hope to inspire you with the unexpected beauty—and mystery—inherent in computation. Again, no mathematics required.

三个基本思想中最重要但鲜为人知的是本书的基本主题：你可以在波和比特之间来回穿梭——在模拟世界和数字世界之间。这个想法只能追溯到 1933 年，当时俄罗斯人 Vladimir Kotelnikov 建立了我们今天所知的它。它的正式名称是采样定理。整本书——作为像素的传记，像素是我们对视觉世界样本的名称——都是关于采样的。像素是代表可见波的不可见位。我的热切意图是让您了解这件魔法，并对它的工作原理感到惊讶。这里也不需要数学。

The most important but least known of the three fundamental ideas is the underlying theme of this book: you can pass back and forth between waves and bits—between the analog and digital worlds. The idea dates back only to 1933 when the Russian Vladimir Kotelnikov established it as we know it today. Its formal name is the Sampling Theorem. This entire book—being a biography of the pixel, with pixel being our name for a sample of the visual world—is about sampling. Pixels are invisible bits that represent visible waves. My fervent intent is that you understand this piece of magic and be amazed by how it works. No mathematics is required here either.

既然我已经在两页中说过三次不需要数学，你可能会想：但是如果我们中的一些人关心数学怎么办？对于您——但实际上对于我所有的读者——我在http://alvyray.com/DigitalLight上提供了一个在线注释站点。在那里，您将找到有关人物、地点和事件的更多详细信息，这些细节会使这本书过于笨拙，无法在其物理封面之间进行放置——您还将找到支持 Digital Light 魔力的数学方程式和使之成为可能的像素. 3

Now that I’ve said no math required three times in two pages, you might be thinking: But what if some of us care about the math? For you—but really for all my readers—I provide an online annotations site at http://alvyray.com/DigitalLight. There you will find additional details about people, places, and events that would have made this book too unwieldy to fit between its physical covers—and you will find as well mathematical equations to support the magic of Digital Light and the pixels that make it possible.3

一个常见的误解是像素是一个小方块颜色。但事实上，像素是一个深刻而抽象的概念，将我们现代媒体世界联系在一起。这是Digital Light的组织原理。

There’s a common misconception that a pixel is a little square of color. But in fact, the pixel is a profound and abstract concept that binds our modern media world together. It’s the organizing principle of Digital Light.

一个视觉场景由无数个颜色点组成。根据定义，无限是太大而无法处理的。那么，我们如何才能用有限数量的离散位（像素）来替换平滑的视觉场景，而不会丢失它们之间的无限信息呢？采样定理告诉我们如何去做。这是使现代媒体世界运转的秘密。

A visual scene consists of an infinite number of points of color. Infinite is, by definition, too large to deal with. So how can we replace a smooth visual scene with only a finite number of discrete bits—the pixels—and not lose an infinity of information between them? The Sampling Theorem tells us how to do it. It’s the secret that makes the modern media world work.

依赖于傅立叶波的采样几乎是在 1930 年代中期与计算同时创建的。采样遇到了计算并孕育了一个孩子，Digital Light，这本书的主题。

Sampling, which depends on Fourier’s waves, was created almost simultaneously with computation in the mid-1930s. Sampling met computation and conceived a child, Digital Light, the subject of this book.

贡献：两项高科技

Contributions: Two High Technologies

本书的第二部分专门介绍了塑造数码光的两项高科技的历史：计算机和电影。与第一部分一样，我直观地介绍了每项技术，并讲述其发明的历史——揭穿一些常见的神话。真实的故事总是比神话更有趣、更鼓舞人心、更复杂。

Part II of the book is devoted to the history of two high technologies that shaped Digital Light: computers and movies. As in part I, I present each technology intuitively and give the history of its invention—debunking some common myths. The true stories are invariably more intriguing, inspiring, and complicated than the myths.

在 Digital Light 中，我们可以从现实世界中获取像素——例如，从国际空间站上的摄像头跟踪最新的飓风。但更重要的是，对于这本书，我们还可以从头开始制作像素。这就是计算机的用武之地，也是我如此详细地介绍高科技发展的原因。

In Digital Light we can take pixels from the real world—from cameras on the International Space Station, say, tracking the latest hurricane. But more importantly for this book we can also make pixels from scratch. That’s where computers come in and why I present the development of that high technology in such detail.

我为这本书所做的研究中的许多惊喜之一是发现第一个像素出现在第一台计算机上。他们是一起出生的。因此，通过确定哪些计算机是世界上第一台计算机，我们还可以了解谁拥有世界上第一台像素。这就是为什么第二部分中关于计算机的章节被称为“数字光的黎明：加速”的原因。它展示了 1947 年用像素制作的第一张图片。它引入了一个巨大力量的驱动概念，称为摩尔定律：

One of many surprises from my research for this book was the discovery that the first pixels occurred on the first computers. They were born together. So by establishing which computers were the first in the world, we also learn who had the first pixels in the world. That’s why the chapter on computers in part II is called “Dawn of Digital Light: The Quickening.” It features the first picture made with pixels, in 1947. And it introduces a driving concept of enormous force, called Moore’s Law:

计算机的任何优点每五年就会提高一个数量级。

Anything good about computers gets better by an order of magnitude every five years.

尽管它很简单，但这种说法是革命性的。1965 年戈登摩尔宣布它时的 1 倍，现在约为 1000 亿，到 2025 年将达到 1 万亿。这是一颗爆炸的超新星。摩尔定律是 50 多年来所有计算机发展背后的强大动力，包括 Digital Light。

Despite its simplicity, that statement is revolutionary. What was a factor of 1 in 1965, when Gordon Moore announced it, is about 100 billion now, and will reach 1 trillion by 2025. It’s an exploding supernova. Moore’s Law is the astonishingly powerful dynamo behind every development of computers for over 50 years—including Digital Light.

Digital Light的数字电影部分来源于经典电影。标题为“电影和动画：采样时间”的章节回顾了这种前数字运动图像技术。它还有助于进一步说明采样：我们熟悉的闪烁胶片“帧”实际上是一个示例。

The digital movie part of Digital Light derives from classic film. The chapter titled “Movies and Animation: Sampling Time” reviews this pre-digital moving-image technology. It also helps to further illustrate sampling: the familiar “frame” of flickering film is actually a sample.

动词take和make 也适用于电影。我们用相机从现实世界的场景中拍摄经典电影。我们根据非现实世界的图画制作经典动画电影。这两种电影的核心奥秘在于它们为什么应该有效。一系列静止帧如何传递运动和情感？至少，采样定理有助于解释运动。第一部数字电影——例如皮克斯的《玩具总动员》 ——是经典动画电影的数字继承者。

The verbs take and make apply to film movies too. We take a classic film movie from real-world sets with a camera. We make a classic animated movie from drawings of a non-real world. The central mystery of both kinds of movies is why they should work at all. How does a sequence of still frames deliver both motion and emotion? The Sampling Theorem helps explain motion, at least. The first digital movies—such as Pixar’s Toy Story—are digital heirs to classic animated films.

数码光的兴起和闪耀

The Rise and Shine of Digital Light

Digital Light 的故事太大，无法在一本书中涵盖，因此必须重点关注这一点。它涵盖了从世纪中叶的第一个像素到千禧年的第一部数字电影的数字光。不出所料，我选择写我从个人经验中最了解的特定技术、人物和历史。我出生在计算机和像素之前，我的职业生涯主要致力于创造第一部数字电影。但是选择的路径可以帮助解释数字光是如何变得更普遍的。电子游戏和虚拟现实并没有那么遥远。

The Digital Light story is too large to cover in a single book, so this one necessarily has to focus. It covers Digital Light from the first pixels at midcentury to the first digital movies at the millennium. Unsurprisingly, I have chosen to write about the particular technologies, people, and history I know best from personal experience. I was born before computers—and pixels—and my career was principally devoted to creating the first digital movies. But that chosen path can help explain how Digital Light came to be more generally. Videogames and virtual reality aren’t that far removed.

第三部分的三章讲述了数码光在黎明后的故事（在第 4 章中讲述）和那些第一个世纪中叶的像素。1965 年的摩尔定律将它们清晰地分为摩尔定律之前的数字光时代 1 和之后的时代 2。标题为“即将到来的事物的形状”的章节专门介绍了第 1 纪元。另外两章“意义的阴影”和“千年与电影”涵盖了第 2 纪元中摩尔定律带来的巨大变化。

The three chapters of part III pick up the story of Digital Light after its dawn (told in chapter 4) and those first midcentury pixels. Moore’s Law of 1965 cleanly separates them into Epoch 1 of Digital Light, before Moore’s Law, and Epoch 2, after. The chapter titled “Shapes of Things to Come” is devoted to Epoch 1. The other two chapters, “Shades of Meaning” and “The Millennium and The Movie,” cover the massive changes wrought by Moore’s Law in Epoch 2.

时代 1 是计算机的巨大时代，但速度非常慢。一些幸运的人可以接触到那些昂贵的野兽。从这一时期开始出现计算机图形学的中心法则：在计算机内部使用三维欧几里得几何和牛顿物理学来描述一个虚构的世界。然后它被一个虚拟相机观察到，它把它的世界观渲染成文艺复兴视角的二维显示。

Epoch 1 was the era of monstrous, but ponderously slow, computers. A few lucky individuals got access to those expensive beasts. From this period came the Central Dogma of computer graphics: a fictitious world is described inside a computer with three-dimensional Euclidean geometry and Newtonian physics. Then it’s observed by a virtual camera that renders its view of the world into two dimensions in Renaissance perspective for display.

我或多或少地涵盖了 Epoch 2 直到 2000 年。千禧年的高潮是皮克斯、梦工厂和蓝天这三个伟大的数字电影工作室的崛起。他们的故事交织在一起。

I cover Epoch 2 until 2000, more or less. The culmination at the millennium was the rise of three great digital movie studios, Pixar, DreamWorks, and Blue Sky. Their stories are intertwined.

我不指望你在读完这本书后就能创作出一部数字电影，但我希望你能明白这是怎么可能的。这就像一堂音乐鉴赏课：学会了音乐原理后，你将无法自己为大提琴谱写巴赫组曲，但你会更好地理解这些组成部分，从而更爱巴赫。了解像《玩具总动员》这样的现代电影是如何制作的可以产生同样的效果。

I don’t expect you to be able to create a digital movie after reading this book, but I hope you might understand how it’s possible. It’s like a music-appreciation course: After you learn the principles of music, you won’t be able yourself to compose a Bach suite for cello, but you’ll better understand the components and so love Bach even more. Understanding just how a modern movie like Toy Story is made can have the same effect.

如何谈高科技

How to Talk about High Technology

从我对本书技术史的研究中得出了几个共同的主题：

Several common themes emerged from my research on the technological histories in this book:

都是音乐

It’s All Music

傅立叶的伟大想法是：世界就是音乐。都是海浪。

Fourier’s great idea was this: The world is music. It’s all waves.

这种音乐洞察力导致了广播，这也许不足为奇。但它也导致了电视。事实上，在它的众多后代中，全是媒体技术——所有最近在大数字融合期间合并的各种媒体。简而言之，傅里叶的伟大思想风靡世界，催生了现代媒体的所有雷电。

This musical insight led to radio, which isn’t a surprise perhaps. But it also led to television. In fact, among its many progeny are all media technologies—all the varied media that recently coalesced during the Great Digital Convergence. In short, Fourier’s great idea stormed the world and fostered all the thunder and lightning of modern media.

但它比这更普遍，远远超出媒体。几乎没有一个科学和技术分支不受它的影响——电学和磁学、光学、X 射线衍射、概率论、地震分析和量子力学。名单还在继续。毫不夸张地说，傅立叶改变了我们对世界的理解。

But it’s even more pervasive than that, reaching far beyond just media. Hardly a branch of science and technology is untouched by it—electricity and magnetism, optics, X-ray diffraction, probability theory, earthquake analysis, and quantum mechanics. The list goes on and on. It’s no exaggeration to say that Fourier changed how we understand the world.

无论您坐在文化过道的哪一边，您都可能认识艾萨克·牛顿和阿尔伯特·爱因斯坦。世界在他们的有生之年认可了牛顿的万有引力理论和爱因斯坦的相对论。但在过去的 200 年里，只有物理学家和工程师才能让傅里叶的火焰保持活力。他们知道傅立叶是家长，并据此庆祝他。他们的工作，在许多不同的领域，展示了傅里叶波浪思想的伟大和普遍性。

Regardless of which side of the culture aisle you sit on, you probably know Isaac Newton and Albert Einstein. The world recognized Newton’s theory of gravitation and Einstein’s theory of relativity in their lifetimes. But for the past 200 years, only physicists and engineers have kept Fourier’s flame alive. They knew that Fourier was paterfamilias and celebrated him accordingly. And their work, in many diverse fields, demonstrated the greatness and universality of Fourier’s wave idea.

傅立叶本人只迈出了关键的第一步。他是第一个以数学方式提出这个想法并通过实验对其进行测试的人。尽管他提出了几乎所有科学领域都会产生数千种解决方案的解释，但他自己只培育了第一朵花，即他对固体热流的解决方案。

Fourier himself took only the crucial first step. He was the first to formulate the idea mathematically and test it experimentally. Although he planted the explanation from which thousands of solutions would grow in nearly all fields of science, he himself cultivated only the first flower, his solution to heat flow in solids.

这种不浪漫的专业是他延迟认可的一个很好的理由。他因其关于热如何流动的理论而成为“科学院著名的傅立叶”。它不像爱因斯坦关于引力是时空扭曲的概念那样充满诗意。

That unromantic specialty is a good reason for his delayed recognition. He was the “celebrated Fourier at the Academy of Science” for his theory of how heat flows. It doesn’t ring of poetry like Einstein’s concept that gravity is a warp in spacetime.

然而，傅立叶的伟大想法对于现代经验来说远比爱因斯坦的更重要。以它的音乐形式来说，它就像时空扭曲一样可爱——而且更直观。它没有理由隐藏在它通常难以理解的数学外衣后面。

Yet Fourier’s great idea is far more fundamental to the modern experience than Einstein’s. Stated in its musical form, it’s as lovely as spacetime warps—and more intuitive. There’s no reason for it to remain hidden behind its usual cloak of impenetrable mathematics.

是时候扭转雨果的评价，​​庆祝这个人和他的好主意了。无处不在的现代数字光技术是最终给予傅立叶应得的正确工具。

It’s high time to reverse Hugo’s assessment and celebrate both the man and his great idea. The ubiquitous modern technology of Digital Light is the right vehicle to finally give Fourier his due.

对不朽的渴望

A Lust for Immortality

让·约瑟夫·傅立叶于 1768 年 3 月 21 日出生于巴黎东南约一百英里的古老省会欧塞尔。十年之内，他的父母死了，留下他和他的十四个兄弟姐妹成为孤儿。革命风起云涌。这个新的美国国家只有一岁，本杰明富兰克林在巴黎用他的浣熊皮帽和调情的方式吸引了他们。3

Jean Joseph Fourier was born on March 21, 1768, in Auxerre, an ancient provincial capital about a hundred miles southeast of Paris. Within ten years both his parents were dead, leaving him and his fourteen siblings orphaned. Revolution was in the air. The new American country was only a year old and Benjamin Franklin was charming them in Paris with his coonskin cap and flirtatious ways.3

孤儿傅立叶有一些特别之处，欧塞尔的好人确保这个有才华的孩子受到教育。他们把他安置在一所由约瑟夫·帕莱（Joseph Pallais）经营的学校里，帕莱以曾是让-雅克·卢梭的音乐老师而闻名。唉，没有证据表明发现世界音乐的人本身就是音乐家。

There was something special about the orphan Fourier, and the good people of Auxerre made sure that the talented child was educated. They placed him in a school run by Joseph Pallais, who was noted for having been Jean-Jacques Rousseau’s music teacher. Alas, there’s no evidence that the man who discovered the music of the world was musical himself.

傅立叶随后在他的家乡欧塞尔参加了皇家军事学院（法国全境有 11 个这样的学院分支机构）。当地支持者再次帮助资助了他。这些军校一致强调科学和数学，傅立叶尤其偏爱数学——疯狂地如此。

Fourier then attended the École royale militaire in his hometown Auxerre (there were 11 such branches of the école in all of France). Local supporters helped subsidize him again. These military schools uniformly stressed science and mathematics, and Fourier especially took to math—maniacally so.

13 岁时，他收集烛台点燃一个大“橱柜”，并将他的数学研究延伸到凌晨，直到“熄灯”。令人窒息的——而且肯定是冰冷的——橱柜损害了他余生的健康。也许那个橱柜首先让他对热产生了特殊的兴趣。

At 13 he collected candle stubs to light a large “cupboard” and extend his study of mathematics into the wee hours, past “lights out.” The suffocating—and surely cold—cupboard compromised his health for the rest of his life. Perhaps that cupboard first gave him his peculiar interest in heat.

但橱柜里的烛光课外训练很快就得到了回报。他获得了数学学生奖。这项技能开启了他的科学生涯，并最终使他不朽。然而，他还获得了修辞方面的一等奖，使他的政治生活成为可能，并直接导致他早日与死亡擦肩而过。在数学技能有机会发展之前，他的口语技巧几乎要了他的命。

But the candlelit extracurricular training in the cupboard soon paid off. He won a student prize in math. This skill launched his scientific career and ultimately made him immortal. And yet he also won a first prize in rhetoric, enabling his political life and leading directly to an early brush with mortality. His speaking skill almost killed him before the mathematical skill had a chance to develop.

危险一开始并不明显。进一步的军校训练可能直接将他带入了军队，但他肯定不是军人类型。一方面，他病得很重，而且是个数学怪胎。于是，在皇家军事学院完成学业后，他进入了教堂。他成为欧塞尔修道院的一名新手，并在那里向其他新手教授数学。大约在这个时候，他将自己的名字神圣化给了让·巴蒂斯特·约瑟夫·傅立叶，这是他此后使用的形式。

The danger wasn’t obvious at first. Further military school training might have led him directly to the army, but he certainly wasn’t the military type. He was sickly for one thing, and a math geek. So, after completing his studies at the École royale militaire, he entered the church. He became a novice at an abbey in Auxerre and taught mathematics to the other novices there. He sanctified his name at about this time to Jean Baptiste Joseph Fourier, the form that he used thereafter.

傅立叶在法国大革命前夕消失在修道院中。他幸存下来的几封信表明他隐约意识到这一点，但无动于衷。他更担心自己的名声和代数论文的状况，而不是法国的状况。“昨天是我的 21 岁生日。在那个年纪，”他在 1789 年 3 月的一封信中苦恼地说，“牛顿已经获得了许多不朽的主张。”

Fourier disappeared into the cloisters on the very eve of the French Revolution. The few letters of his that survive show he was vaguely aware of it, but indifferent. He was more worried about his fame and the state of an algebra paper than the state of France. “Yesterday was my 21st birthday. At that age,” he anguished in a March 1789 letter, “Newton had acquired many claims to immortality.”

9 月，他又写了一封信，哀叹代数论文的命运。在这两封信之间，革命已经开始。但 9 月的信中没有提到那些动荡的事件。

In September he wrote another letter, bemoaning the fate of his algebra paper. In the time between the two letters, the Revolution had begun. But the September letter said nothing about those tumultuous events.

然而，傅里叶的私人世界在那之后确实开始发生变化。他于 12 月在巴黎科学院提交了一篇关于“代数方程”的论文——大概是让他非常担心的那篇论文。

Nevertheless, Fourier’s private world did begin to change after that. He presented a paper on “algebraic equations”—presumably the paper that had worried him so much—before the Academy of Science in Paris in December.

他没有宣誓就离开了修道院。无论如何，革命政府很快就镇压了宗教秩序。

And he departed the abbey without taking his vows. The revolutionary government soon suppressed religious orders anyway.

然而，傅立叶在接下来的三年里并不是革命者。他改为在欧塞尔教数学。特别是，他没有签署欧塞尔革命人民协会向巴黎国民公会要求审判国王路易十六的请愿书。

Yet Fourier wasn’t a revolutionary for the next three long years. He taught math in Auxerre instead. In particular, he didn’t sign the petition from the revolutionary Popular Society of Auxerre to the National Convention in Paris demanding the trial of King Louis XVI.

但在 1793 年初，就在国王被斩首一个月后，我们开始听到公民傅立叶的来信。

But in early 1793, just a month after the king’s beheading, we begin to hear from Citizen Fourier.

众所周知，法国大革命的黎明令华兹华斯着迷，而年轻的傅立叶在他迟来的拥抱它后，在同一个天堂中欢欣鼓舞。他的话比诗人的更笨拙：“可以设想在我们中间建立一个不受国王和教士影响的自由政府的崇高希望。” 但他同样充满热情：“我很容易就迷上了这项事业，在我看来，这是任何国家所从事的最伟大、最美丽的事业。” 5

The dawn of the French Revolution famously enthralled Wordsworth, and young Fourier exulted in that very same heaven once he belatedly embraced it. His words were clumsier than the poet’s: “It was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests.” But he was just as passionate: “I readily became enamored of this cause, in my opinion the greatest and the most beautiful which any nation has ever undertaken.”5

傅立叶不必从这些情绪跳到政治本身。1793 年 2 月，他在欧塞尔的革命公社发表了激动人心的首次演讲。他有一个为共和国军队招募当地新兵的计划。欧塞尔大众社会给他留下了深刻的印象，采纳了他的计划并邀请他加入。恐怖统治现在已经全面展开——最终有一万多人被斩首，被怀疑是国家的敌人。他明智地接受了“邀请”。

Fourier didn’t have to leap far from these sentiments to politics itself. He gave a rousing debut speech in February 1793 to the revolutionary commune in Auxerre. He had a plan for raising local recruits for the Army of the Republic. Favorably impressed, the Popular Society of Auxerre adopted his plan and invited him to join. The Reign of Terror was now in full play—with ultimately over ten thousand beheadings of suspected enemies of the state. He wisely accepted the “invitation.”

但天真的傅立叶——最近的新手——立刻就犯了错误，他的时机已经很糟糕了。他那受过修辞训练的舌头让他陷入了严重的麻烦。他不明智地用它来保护奥尔良的三个公民。不明智，因为他们已经在罗伯斯庇尔的敌人名单上，而罗伯斯庇尔是恐怖分子的“沙皇”。

But naive Fourier—the recent novice—immediately blundered, and his timing was about as bad as could be. That rhetoric-trained tongue of his got him into serious trouble. Unwisely he used it to defend three citizens of Orléans. Unwise because they were already on Robespierre’s enemies list, and Robespierre was the Terror’s “tsar.”

革命者迅速解除了傅立叶在欧塞尔以外的所有职责。由于无法进一步推进共和国，他苦苦挣扎，跋涉到巴黎，亲自会见罗伯斯庇尔为他的案子辩护。这种大胆的策略适得其反。他对奥尔良囚犯的支持使他在罗伯斯庇尔的敌人短名单上占有一席之地。由于革命的扭曲逻辑，尽管欧塞尔一直支持的公民提出抗议，但傅里叶所倡导的恐怖行动还是在 1794 年 7 月 17 日将他送进了监狱。这实际上是死刑。

The revolutionaries promptly relieved Fourier of all duties outside Auxerre. Miserable at not being able to further the Republic, he trekked to Paris and met with Robespierre himself to plead his case. This bold tactic seriously backfired. His support for the Orléans prisoners had earned him a spot on Robespierre’s short list of enemies. With the contorted logic of the Revolution, and despite protests from the always supportive citizens of Auxerre, the very Terror that Fourier had championed threw him into jail on July 17, 1794. It was effectively a death sentence.

“我经历了各种程度的迫害和不幸，”他说。“我的对手没有一个比这更危险，我是我们同胞中唯一被判处死刑的人。” 6

“I experienced every degree of persecution and misfortune,” he said. “None of my adversaries have run more dangers, and I am the only one of our compatriots who was condemned to death.”6

几天之内，下一站将是巴黎的革命法庭和断头台上的橡皮图章。他有理由感到害怕。他不可能知道仅仅十天之后——7 月 27 日，或者 9 日，法国大革命时间——罗伯斯庇尔会重重摔倒。那个叫人头倒地，从来没有受够——再次引导华兹华斯的人——亲自尝过断头台。幸运的是科学的未来，特别是像素，罗伯斯庇尔的脑袋拯救了傅里叶的脑袋。7

The next stop, within days, would’ve been the Revolutionary Tribunal in Paris and a rubberstamp to the guillotine. He was justifiably terrified. He couldn’t have known that just ten days later—July 27, or 9 Thermidor, French Revolutionary Time—Robespierre would fall hard. He who bade heads fall and never had enough—to channel Wordsworth again—tasted the guillotine himself. Lucky for the future of science, and the pixel in particular, Robespierre’s head saved Fourier’s.7

海浪

The Wave

1789 年的那篇代数论文——将革命从傅立叶的脑海中推开的那篇论文——是这位伟大科学家的第一次公开瞥见吗？它是否包含他伟大想法的关键？正如数学家所说，这无疑提高了他的数学技能并提高了他的“数学成熟度”。但我们不知道里面有什么。

Was that algebra paper of 1789—the one that pushed the Revolution from Fourier’s mind—the first public glimpse of the great scientist? Did it contain keys to his great idea? Surely it sharpened his mathematical skills and increased his “mathematical maturity,” as mathematicians say. But we don’t know what was in it.

我们不知道，特别是傅立叶何时开始使用波，这是他伟大创意中的基本形状。但到了 1807 年，他已经公开了它。这是你通过展开一个完美的圆圈得到的形状，所以它是一种革命性的形式。它非常简单。像素具有崇高的传统。

We don’t know, in particular, when Fourier first began to use the wave, the fundamental shape in his great idea. But by 1807 he had gone public with it. It’s the shape you get by unwinding a perfect circle, so it’s a revolutionary form. And it’s elegantly simple. The pixel has a noble heritage.

要获得傅立叶波的图像，请从一个圆圈开始（图 1.1）。老式的模拟钟面可以达到目的。秒针的尖端围绕着那个圆圈稳定地移动，从一个分钟标记到下一个标记，一秒一秒。下图显示了三秒后的上表盘。

To get a picture of Fourier’s wave, start with a circle (figure 1.1). An old-fashioned analog clock face serves the purpose. The tip of its second hand moves steadily around that circle, from a single minute mark to the next one, second by second. The lower picture shows the upper clockface, three seconds later.

较大的红点描绘了波浪。动画电影在这里会特别有启发性，但如果没有这一点，请将时间的流逝想象为向右移动，如时间箭头所示。时钟边缘的每个刻度线都有一个沿着箭头的刻度线。想象一下，红点总是被一条拉紧的水平带拴在秒针的尖端。随着时间的推移，该带向右延长，那个红点所描绘的路径就是波。

The larger, red dot traces out the wave. An animated movie would be especially instructive here, but short of that, imagine the passage of time as movement to the right, as shown by time’s arrow. There’s a tick mark along the arrow for each tick mark along the edge of the clock. Imagine that the red dot is always tethered to the tip of the second hand by a taut horizontal band. The band lengthens toward the right as time passes, and the path traced by that red dot is the wave.

这里的重点是，简单地说，波浪看起来像什么，直观地说，它与圆密切相关。这种亲密关系的细节不如直觉重要，但更多的细节可能有助于更难忘地记录直觉。

The important points here are, simply, what the wave looks like and, intuitively, that it’s intimately related to the circle. The details of that intimate relation aren’t as important as the intuition, but a few more details might help to register the intuition more memorably.

考虑时钟的中心线——连接 9 点钟和 3 点钟的线。红点始终标记秒针尖端在该中心线上方或下方的当前高度。在为上图选择的（纯粹是任意的）时刻，自记录的第一个位置以来，红点已经追踪了 23 个位置，因为从那时起已经过去了 23 秒。在下一个滴答声中，红点将向右移动到波浪上的下一个点。之后再打两个刻度，得到下图。因此，当秒针在时钟上一次又一次地滴答作响时，它的尖端——实际上是附着在尖端的红点——描绘出所示的波浪形路径，上下，上下。. .

Consider the centerline of the clock—the line that connects 9 o’clock to 3 o’clock. The red dot always marks the current height of the tip of the second hand above or below that centerline. At the (purely arbitrary) moment chosen for the upper illustration, the red dot has traced through twenty-three positions since the first position recorded, because twenty-three seconds have elapsed since then. During the next tick the red dot will move right to the next dot on the wave. And two more ticks after that yields the lower illustration. So as the second hand ticks its way around the clock, again and again, its tip—actually the red dot attached to the tip—traces out the wavy path shown, up and down, up and down . . .

秒针每分钟转一圈，所以红点所描绘的波浪永远向右移动。它也永远向左延伸。图中的波浪似乎在中午后一分钟开始——另一个纯粹的任意选择——但显然时钟一直在滴答作响，直到你可能想追踪的时间。

The second hand makes one revolution after another every minute, so the wave traced by the red dot heads to the right forever. It extends to the left forever too. The wave in the figure appears to start one minute after noon—another purely arbitrary choice—but obviously the clock’s been ticking off the seconds as far back as you might care to trace.

图中的波是傅里叶波之一。他没有发明它们，但他深刻地利用了它们。这些是他音乐的组成部分。数学家将这种特别可爱的圆形波称为正弦波。由于这是我们需要的唯一一种 wave，我通常将其简称为wave。

The wave in the picture is one of Fourier’s waves. He didn’t invent them, but he made profound use of them. These are the components of his music. Mathematicians call this particularly lovely circle-made wave a sine wave. Since this is the only kind of wave we need, I’ll usually refer to it simply as a wave.

关于傅立叶波的一切都是简单、美丽、优美、完美的。科学家和工程师对傅立叶的热流解以及他伟大的音乐理念有一个正式的术语。他们将其描述为谐波。

Everything about Fourier’s wave is simple, beautiful, graceful, perfect. Scientists and engineers have a formal term for Fourier’s solution for heat flow, and for his great musical idea generally. They describe it as harmonic.

那个波浪形无处不在，并且一直伴随着你。从您家中或办公室的任何插座流出的电流电压都是波。由于其波动性，它被称为交流电，AC。它通常源于转子的转动，例如大坝后面的水。转子的圆圈展开成交流电或波电流。电机使转子向右转动。波形电流进入电机并将能量卷回圆形机械运动。电风扇吸进一波，又放出一个圆圈。

That wave shape is everywhere, and with you all the time. The voltage of the electrical current coming out of any outlet in your home or office is a wave. It’s called alternating current, AC, because of its wave nature. It often originates with the turning of a rotor, by water behind a dam for example. The circle of the rotor unfurls into AC, or wave, current. A motor turns that rotor right around. The wave-shaped current enters a motor and furls the energy back into circular mechanical motion. An electric fan takes a wave in and puts a circle out.

另一个熟悉的正弦波来自媒体世界。这是您最喜欢的 FM 广播电台的电话号码。我在旧金山地区的 KCSM 爵士乐是 91.1。数字 91.1 正式指定了当局分配给 KCSM 的波，仅用于广播。它描述了电台用来将音乐传递给听众的电磁波。

Another familiar sine wave comes from the media world. It’s the call number of your favorite FM radio station. Mine is 91.1 for KCSM jazz in the San Francisco area. The number 91.1 officially designates the wave assigned by authorities to KCSM for its sole use in broadcasting. It describes the electromagnetic wave that station uses to carry its music to listeners.

尽管所有的傅立叶波都具有相同的形状，但它们在两个方面有所不同——它们摆动的速度（称为频率）和波峰的高度（振幅）。在时钟图中，秒针波峰的频率是多少？它每分钟出现一次波峰，这就是它的自然频率——每分钟一个完整的波周期。循环是完全正确的词。当秒针绕钟面转一圈时，附在其尖端的红点会及时描绘出它的一个完整周期。绕圆周一圈产生一个波周期。

Although all Fourier’s waves have the same shape, they differ in two ways—how fast they wiggle (called frequency) and how high they crest (amplitude). In the clock figure, how frequently does the second-hand wave crest? It crests once every minute, so that’s its natural frequency—one complete wave cycle per minute. Cycle is exactly the right word. As the second hand cycles around the clock face once, the red dot attached to its tip paints out one complete cycle of its wave in time. One revolution around the circle yields one wave cycle.

时钟的分针尖端勾勒出一模一样的平滑曲线，但速度更慢。分针每小时只有一次波峰。它的频率是每小时一个周期，比秒针慢六十倍。第三个是时针波。它具有三者中最低的频率，每半天一个周期。

The tip of the clock’s minute hand traces out a curve of exactly the same smooth shape, but more slowly. The minute-hand wave crests only once per hour. Its frequency is one cycle per hour, sixty times slower than the second hand. A third one is the hour-hand wave. It has the lowest frequency of the three, one cycle per half day.

您在 KCSM 的电话号码中也遇到过频率；91.1 是该无线电台使用的波的频率（以每秒数百万个周期为单位）。那些无处不在的交流电源插座每个都以每秒 60 个周期的频率发出波（在美国）。

You’ve also met frequency in KCSM’s call number; 91.1 is the frequency (in millions of cycles per second) of the wave used by that radio station. And those ubiquitous AC electrical outlets each deliver a wave with a frequency of 60 cycles per second (in the US).

秒针、分针和时针波的幅度和频率各不相同。我把分针画得比秒针略短，所以分针的波峰略低。由于波峰的最大高度是其振幅，因此分针波的振幅低于秒针波的振幅。时针更短，所以时针波的振幅是三者中最低的。

The second-, minute-, and hour-hand waves differ in amplitude as well as frequency. I drew the minute hand slightly shorter than the second hand, so the wave crests for the minute hand are slightly lower. Since the maximum height of a wave’s crest is its amplitude, the amplitude of the minute-hand wave is lower than that of the second-hand wave. The hour hand is even shorter, so the hour-hand wave’s amplitude is the lowest of the three.

对于傅立叶的目的，波可以有任何频率和任何幅度，只要它是一个正弦波——一个展开的圆。时钟图（图 1.1）产生了三个这样的波，图 1.2 显示了另外三个，它们都具有优美的形状，并且仅在摆动速度和高度（频率和幅度）方面彼此不同。他们都有形状相同，意思是所有三角形都具有相同的形状。三角形只要有三个直边就可以成为三角形，而波浪只要是展开的圆形就可以成为波浪。

For Fourier’s purposes a wave can have any frequency and any amplitude so long as it’s a sine wave—an unfurled circle. The clock figure (figure 1.1) yielded three such waves, and figure 1.2 shows three more, all gracefully shaped and differing from one another only in wiggle speed and height—frequency and amplitude. They all have the same shape, in the sense that all triangles have the same shape. A triangle only has to have three straight sides to be a triangle, and a wave has only to be an unfurled circle to be a wave.

注意这张图片中波浪的另一件事：它们周期中的不同点与图片的左边缘对齐。顶部波的波峰与边缘对齐，中间的波谷，以及底部波之间的某个位置。如果你向左或向右滑动任何波，它的频率和幅度不会改变，但它相对于另一个波的位置会改变。这很重要，因为傅立叶让我们将波加在一起。如果波的排列方式不同，您会得到不同的总和。

Notice one other thing about the waves in this picture: different points in their cycles are aligned with the left edge of the picture. The crest of the top wave aligns with the edge, the trough of the middle one, and somewhere in between for the bottom wave. If you slide any wave left or right, its frequency and amplitude don’t change, but its position relative to another wave does. This is important because Fourier has us add waves together. You get different sums if the waves are aligned differently.

我们使用阶段这个词来描述波浪的位置。著名的月相告诉我们月亮在其周期中的位置。它是满月或新月，或者是在满月和新月之间的盈亏。波是循环的，所以它也有一个相位。在图 1.2 中，顶部的波浪在图片的左边缘处于波峰（满月），而中间的波在该处处于低谷（新月）。底部的刚刚开始从它的顶峰减弱到它的谷底。改变波的相位会使整个波向左或向右移动。但请注意，如果您将其移动一个完整的周期，则该波正是原始波。没有办法区分它们。因此，在单个周期中指定一个位置（即相位是什么）就足以指定整个波的位置。对于傅立叶的目的，波可以处于任何相位。

We use the word phase to describe the position of a wave. The moon’s phases famously tell us where the moon is in its cycle. It’s a full moon or a new one, or it’s waxing or waning between full and new. A wave is cyclic, so it has a phase too. In figure 1.2 the top wave is cresting (full moon) at the left edge of the picture, and the middle wave is troughing (new moon) there. The bottom one is just starting to wane from its full crest to its trough. Changing the phase of a wave shifts the entire wave left or right. But notice that if you shift it by an entire cycle, then the wave is exactly the original wave. There’s no way to tell them apart. So, specifying one place in a single cycle—which is what a phase is—is enough to specify the position of an entire wave. For Fourier’s purposes a wave can be in any phase.

傅立叶的伟大想法现在可以一瞥：世界的大量模式——包括我们能看到或听到的一切，还有很多很多——可以准确地描述为这种类型的波的总和，仅此而已. 傅立叶频率是这种描述中波的频率。他们的和谐是世界的音乐。这个想法是如此之大且违反直觉，以至于可能难以理解。所以让我们从音乐本身开始，这是一种熟悉的物理现实，有助于直观地理解傅立叶的深刻思想。

A first glimpse of Fourier’s great idea is now possible: a vast number of patterns of the world—including everything we can see or hear, but also many, many more—can be described exactly as a sum of waves of this type and nothing else. Fourier’s frequencies are the frequencies of the waves in such a description. Their harmony is the music of the world. This idea is so large and counterintuitive that it may be hard to grasp. So let’s start with music itself, a familiar physical reality that helps make intuitive sense of Fourier’s profound idea.

声音

Sound

音乐是由——而且只是——不同频率的波组成，当然也称为声波。小提琴上的琴弦以不同的频率振动，钢琴也是如此。与时钟每分钟一个周期的秒针频率相比，钢琴的中间 C 是来自以每秒 262 个周期振动的弦的声波的速度恶魔。单簧管或长笛以特定频率共振，管风琴的每个管子也是如此。抒情的花腔女高音歌唱的频率比中音高，也比男中音或低音高得多。我们说女高音唱得更高而不是频率，指的是我们大脑对音乐的感知，而不是音乐创作的物理原理，但它等同于同一件事。正如我们所说，一个和弦本质上是几个波——比如三个或四个——同时演奏，或者加在一起。合唱团由许多不同音高的声音组成，管弦乐队由许多不同频率的乐器组成，从弦乐贝司到短笛。

Music is composed of—and only of—waves of different frequencies, called sound waves of course. The strings on a violin vibrate at different frequencies, and the same for a piano. Compared to a clock’s pokey second-hand frequency of one cycle per minute, a piano’s middle C is a speed demon of a sound wave from a string vibrating at 262 cycles per second. A clarinet or a flute resonates at certain frequencies, as does each of a pipe organ’s pipes. Lyric coloratura sopranos sing at a higher frequency than do altos and much higher than baritones or basses. We say sopranos sing at a higher pitch rather than frequency, referring to our brain’s perception of the music rather than to the physics of its creation, but it amounts to the same thing. A chord is essentially several waves—say three or four—played simultaneously, or added together, as we say. A choir consists of many voices at different pitches, and an orchestra comprises many instruments at different frequencies, from string basses to piccolos.

音乐的动态，从弱音到强音，反映了音乐中波的幅度。振幅越大，声音越大。巨大的双调音管风琴管——脚踏板最大限度地放在地板上——用全能者的可怕力量震撼着一座大教堂。用力敲击钢琴键或调高收音机的音量以增加波的幅度。毫无疑问，收音机或立体声系统的主要部件是其放大器。

The dynamics of music, from pianissimo to fortissimo, reflect the amplitudes of the waves in the music. The higher the amplitude, the louder the sound. The massive double diapason organ pipe—with the foot pedal maxed to the floor—shakes a cathedral with the awfulness of the Almighty. Strike the piano key harder or turn up the volume on a radio to increase the amplitude of the waves. It’s no surprise that the principal component of a radio or a stereo system is its amplifier.

当您描述音乐时，傅立叶的想法似乎很自然，但当您考虑到所有声音（不仅仅是音乐）都是由音频波组成时，他的想法的力量就会开始显现出来。我们谈论低频隆隆声和高音哨声。众所周知，狗能听到比我们更高的音调。我们对傅立叶大思想最本能的直觉是，任何声音或音乐都是由不同频率的声波组成的，它们全部叠加在一起，被我们的耳朵和大脑解读为斯特拉文斯基仪式、心爱的孩子的声音，甚至是建筑工地的噪音。

Fourier’s idea seems only natural when you’re describing music, but the power of his idea starts coming into focus when you consider that all sound—not just music—is composed of audio waves. We speak of low-frequency rumbles and high-pitched whistles. Dogs famously hear higher pitches than we do. Our most visceral intuition of Fourier’s big idea is that any sound or music is composed of audio waves of different frequencies all added together and interpreted by our ears and brain as a Stravinsky rite, a beloved child’s voice, or even construction site noise.

图 1.3 以不同频率和幅度的傅里叶波形式显示了单词yes（时间向右）。左边的“y”部分包含最低频率和最高振幅——它是单词的重音部分。中间的“e”部分具有最低幅度和混合频率。右边的“s”部分具有较低的幅度和最高的频率——“s”的嘶嘶声。

Figure 1.3 shows the word yes as Fourier waves of various frequencies and amplitudes (time proceeds to the right). The “y” part at the left contains the lowest frequencies and the highest amplitudes—it’s the accented part of the word. The “e” part in the middle has lowest amplitudes and mixed frequencies. The “s” part at the right has lower amplitudes and highest frequencies—the hiss of the “s.”

声波实际上是由——它们的“材料”是——空气的有节奏的压缩，或压力波。考虑一下扬声器系统中的低音扬声器——在响亮的低音音调下您实际上可以感觉到振动的大低音扬声器。很容易想象低音喇叭的快速移动膜在它面前晃动着空气。这种脉动的空气从低音扬声器的表面移动——可以说是被它推动的。更大的声音会在每个更大的振动中引起更高的空气压缩。

Sound waves are actually made of—their “stuff” is—rhythmic compressions of air, or pressure waves. Consider the woofer in your speaker system—the big one that you can actually feel vibrate on loud bass tones. It’s easy to imagine the woofer’s fast-moving membrane shaking the air in front of it. This pulsing air moves from the woofer’s surface—pushed along by it, so to speak. A louder sound causes higher compression of the air in each, larger vibration.

我们直接了解这些压力波。想想一个低速骑手在洛杉矶的克伦肖大道上慢慢地驼背，巨大的音箱摇晃着附近的窗玻璃。

We understand these pressure waves directly. Think of a lowrider humping slowly down Crenshaw Boulevard in Los Angeles with its massive boombox shaking nearby windowpanes.

一个非常响亮和低沉的声波将这些高压波转换为物理振动，似乎是地球本身的振动。但实际上，那些惊天动地的波浪，与进入我们耳朵，使我们的耳膜随着低音喇叭、音箱或双深管风琴一起振动的波浪完全一样。然后一个聪明的小骨头系统——令人愉快的名字是锤子、铁砧和马镫——将这些振动传递到内耳，在那里成千上万的微小毛细胞对不同的频率做出反应。它们将频率信息直接传递到大脑。

A very loud and low sound wave converts those high-pressure waves into physical vibrations, seemingly of the earth itself. But actually, those earth-shaking waves are exactly the same as the waves that enter our ears and make our eardrums vibrate in unison with the woofer, boombox, or double deep organ pipe. Then a clever system of little bones—the delightfully named hammer, anvil, and stirrup—pass these vibrations to the inner ear where thousands of tiny hair cells respond to the different frequencies. They pass the frequency information directly into the brain.

正常的年轻人耳可以听到每秒 20 次循环到每秒 2 万次循环之间的所有频率。现代用法将“每秒周期数”缩短为“赫兹”（缩写为 Hz），但我会坚持使用较长的短语以保持其直观性。还有人耳无法听到的其他声音，比如狗能听到但我们听不到的超声波哨子。

The normal young human ear can hear all frequencies between 20 cycles per second and 20 thousand cycles per second. Modern usage shortens “cycles per second” to “Hertz” (abbreviated Hz), but I’ll stick with the longer phrase to preserve its intuitiveness. There are other sounds outside the human ear’s capability, like the ultrasonic whistle the dog can hear but we cannot.

但是视觉中的波是什么——导致像素的波？视觉振动的频率是多少？

But what are the waves in vision—the ones that lead to pixels? At what frequencies does vision vibrate?

拿破仑·波拿巴

Napoleon Bonaparte

傅立叶出生仅一年多后，Nabolione Buonaparte 就出生在科西嘉岛，科西嘉岛与巴黎和欧塞尔接壤，但位于法国大陆东南约一百英里处（图 1.4）。科西嘉岛在这两个男孩出生的间隙已经成为法国人，所以波拿巴只是勉强是法国人。他的名字在他 20 多岁时才流行起来，当时他将其归结为拿破仑·波拿巴。8

Just over a year after Fourier’s birth, Nabolione Buonaparte was born on Corsica, an island in line with Paris and Auxerre but about a hundred miles southeast of mainland France (figure 1.4). Corsica had become French in the interval between the births of the two boys, so Buonaparte was just barely French. His name caught up only later, in his twenties, when he Frankified it to Napoleon Bonaparte.8

波拿巴参加了布赖耶纳城堡的皇家军事学院。这与傅立叶就读的学校系统相同，他在大约 60 英里外的欧塞尔就读。因此，波拿巴接受了与傅立叶基本相同的科学和数学训练。这足以让波拿巴对数学产生持久的兴趣。他他一生都与数学家互动，而不仅仅是傅立叶。甚至还有一个几何定理，拿破仑定理，以他的名字命名。

Bonaparte attended the École royale militaire in Brienne-le-Château. This was in the same system of schools attended by Fourier, who was at the one in Auxerre about 60 miles away. So Bonaparte received essentially the same training in science and mathematics as Fourier. It was enough to give Bonaparte an abiding interest in math. He was to interact with mathematicians all his life, and not just Fourier. There’s even a geometry theorem, Napoleon’s Theorem, named for him.

波拿巴在巴黎的精英军事学院继续深造，深入军事机构。他在那里的训练细节与我们无关，但他的期末考试却很重要。他的考官形容波拿巴“对数学有透彻的了解”。他是皮埃尔-西蒙·拉普拉斯，有时被称为法国的艾萨克·牛顿。

Bonaparte moved deeper into the military establishment with further schooling at the elite École militaire in Paris. Details of his training there don’t concern us, but his final exams do. His examiner described Bonaparte as having “a thorough knowledge of mathematics.” He was Pierre-Simon Laplace, sometimes called France’s Isaac Newton.

波拿巴后来让拉普拉斯成为法国参议院议员，此举令当代美国人感到难以置信。一位著名的数学家和物理学家担任参议员！

In a move that boggles the mind of a contemporary American, Bonaparte was later to make Laplace a member of the French Senate. A noted mathematician and physicist as senator!

傅里叶遇见波拿巴

Fourier Meets Bonaparte

罗伯斯庇尔倒台后，革命的法国政府不仅将傅立叶从监狱中释放出来，还授予他在新成立的巴黎综合理工学院（现为法国麻省理工学院和加州理工学院合并）的教授职位。最后，他在巴黎自称数学，并遇到了它的领军人物，比如拉普拉斯。这使他很好地吸引了雄心勃勃的波拿巴​​的注意，他正在寻找“专家”陪伴他去埃及旅行。并在身体上和政治上获得更多热量。

After Robespierre’s fall, the revolutionary French government not only freed Fourier from jail but awarded him a plum job as professor at the newly formed École polytechnique in Paris—now France’s MIT and Caltech combined. Finally, he was professing mathematics in Paris and meeting its leading lights, such as Laplace. This positioned him nicely to catch the eye of the ambitious Bonaparte who was looking for “savants” to accompany him on a trip to Egypt. And to catch a lot more heat—physically and politically.

波拿巴最近在与奥地利人的战斗中征服了意大利，并以英雄的身份回到了巴黎。有了这样的赞誉和一支军队，法国政府正确地将他视为威胁。看到他（更不用说他的军队）于 1798 年启程前往埃及并进行新的征服，他们并没有不高兴。这次离开最终使傅立叶和波拿巴走到了一起。

Bonaparte had recently conquered Italy in a fight against the Austrians and returned to Paris a hero. With such acclaim, and an army in tow, the French government rightly perceived him as a threat. They were not unhappy to see him (never mind his army) set off in 1798 for Egypt and a fresh conquest. This departure finally brought Fourier and Bonaparte together.

为了效仿他的英雄亚历山大大帝，波拿巴率领一队法国知识分子（他的学者）参加了入侵，其中包括来自巴黎综合理工学院的年轻教授傅立叶。同样，与我们时代的对比是惊人的。想象一下，在第一架飞往伊拉克或阿富汗的飞机上，一支由领先的数学家和考古学家组成的团队。

In emulation of his hero Alexander the Great, Bonaparte took a team of French intellectuals (his savants) on the invasion, including the young professor Fourier from the École polytechnique. Again, the contrast to our times is striking. Imagine a team of leading mathematicians and archaeologists on the first plane into Iraq or Afghanistan.

埃及探险队在军事上失败了，但在智力上却取得了飞速的成功。罗塞塔石碑的发现是一个著名的高点。埃及学就是从这些开始的。波拿巴本人在那里创立了埃及研究所并担任副校长。傅立叶很快就成为了它的常任秘书。在接下来的十年左右的时间里，他为埃及运动的持久学术声誉做出了贡献，他撰写了大量（20 多卷） 《埃及描述》的导言——对波拿巴过度赞扬。波拿巴甚至在这里和那里进行了一些编辑，以试图从军事失败中夺取知识分子的胜利。

The Egyptian expedition was a military flop, but a soaring intellectual success. The discovery of the Rosetta Stone was a famous high point. Egyptology sprang from those beginnings. Bonaparte himself founded the Institut d’Egypte while there and was its vice president. Fourier soon became its permanent secretary. He contributed to the lasting scholarly reputation of the Egyptian campaign by writing, over the next decade or so, the introduction to the massive (twenty-plus volumes) Déscription de l’Egypte—with overly fulsome praise for Bonaparte. Bonaparte even dropped in a few edits here and there in an effort to snatch an intellectual victory from the jaws of military defeat.

法国在亚历山大港、开罗和大金字塔取得了几次初步胜利后，纳尔逊海军上将和英国人在埃及港口摧毁了法国舰队。尽管英国控制了地中海，但波拿巴在埃及呆了一年多一点后，还是设法偷偷穿过它到达了法国，几乎把其他所有人——以及许多未完成的军事任务——都抛在了后面。

After several initial French victories—at Alexandria, Cairo, and the Great Pyramids—Admiral Nelson and the British destroyed the French fleet at harbor in Egypt. Despite British control of the Mediterranean, Bonaparte managed to sneak across it to France, after a little more than a year in Egypt, leaving nearly everybody else—and much unfinished military business—behind.

他的动机是尝试控制法国——他当然做到了。就在 18 世纪末，他成为了新政府的第一任执政官——这是他成为皇帝的第一步。

His motivation was to take a shot at controlling France—which of course he accomplished. He made himself First Consul of a new government just before the end of the eighteenth century—the first step on his march to become emperor.

当他的学者试图跟随他回家时，他们就没那么幸运了。英国人允许他们通过海上封锁，但保留了罗塞塔石碑。它仍然是大英博物馆中参观人数众多的珍品。

When his savants attempted to follow him home, they weren’t so lucky. The British allowed them through the naval blockade but kept the Rosetta Stone. It’s still a highly visited treasure in the British Museum.

波拿巴仓促而投机取巧地离开埃及，让让·巴蒂斯特·克莱伯 (Jean Baptiste Kléber) 陷入了军事混乱。波拿巴任命他为埃及军队的将军，并通过邮件将此事告知他，因此他无法拒绝。波拿巴用这个伎俩赢得了克莱贝尔的蔑视，后者被留下来收拾残局。

Bonaparte’s hasty and opportunistic departure from Egypt left Jean Baptiste Kléber in the lurch with a military mess. Bonaparte made him general in charge of the army in Egypt, informing him of this via mail so he couldn’t refuse. With this trick Bonaparte gained the scorn of Kléber, who was left to mop up.

现在傅立叶犯了他的第二个重大政治错误。他变得与这位不幸的将军过于亲近了。Kléber 任命他为埃及的分局局长。然后，当一名学生在开罗暗杀克莱伯时，傅立叶为这位将军发表了悼词。那个有天赋但很麻烦的舌头再次摇摆不定：首先他得罪了罗伯斯庇尔，现在得罪了波拿巴。

Now Fourier made his second major political mistake. He became too closely identified with the unhappy general. Kléber made him a bureau director in Egypt. Then when a student assassinated Kléber in Cairo, Fourier gave the general’s eulogy. That gifted, but troublesome, tongue was wagging again: first he’d offended Robespierre, now Bonaparte.

波拿巴显然不希望傅立叶在首都提出克莱伯的观点。他不想让法国知道他在埃及战役中不那么高尚的军事细节。傅立叶曾希望，在埃及之后，他可以在巴黎恢复他的声望地位，成为知识分子行动的中心。相反，波拿巴将直言不讳的傅立叶流放到格勒诺布尔。

Bonaparte clearly didn’t want Fourier propounding Kléber’s views in the capital. He didn’t want France to know the less-than-noble military details of his Egyptian campaign. Fourier had hoped that after Egypt he could resume his prestigious position in Paris, at the center of the intellectual action. Instead, Bonaparte exiled the outspoken Fourier to Grenoble.

但在这样做时，他委婉地选择了他的话，“要求”傅立叶担任伊泽尔省的省长，该省由格勒诺布尔管辖。换句话说，波拿巴“要求”他成为一个比巴黎更靠近科西嘉的省的省长（省长）——距离行动还有很长的路要走。

But in doing so, he chose his words euphemistically, “requesting” that Fourier take the job as prefect of the Isère department, governed from Grenoble. In other words, Bonaparte “asked” him to become the provincial governor (prefect) of a province that was closer to Corsica than Paris—a long way from the action.

傅立叶接受了。波拿巴现在是法国最有权势的人。然而，这是流放，这就是傅立叶的看法。在埃及之后的十年里，他是法国唯一一位不在巴黎的领先数学家和物理学家。

Fourier accepted. Bonaparte was now the most powerful man in France. It was exile, however, and that’s how Fourier perceived it. He was the only leading mathematician and physicist in France who wasn’t in Paris in the decade after Egypt.

想象

Vision

您可能很乐意将您听到的内容视为波的总和，但您不会以这种方式思考您看到的内容。那么，下一步将需要一些时间更多解释。傅立叶的伟大想法是视觉世界，就像音乐一样，是由波的总和构成的。但就视觉而言，波是空间波。它们是二维的。看到它们需要一点练习，但是一旦你养成了寻找它们的习惯，这一点也不难。我们会习惯于看到它们，然后实现傅里叶所做的飞跃。

You are probably comfortable thinking about what you hear as a sum of waves, but you don’t think about what you see that way. This next step, then, is going to take a bit more explaining. Fourier’s great idea is that the visual world, just like music, is made of a summation of waves. But in the case of vision, the waves are space waves. They’re two-dimensional. It takes a little practice to see them, but once you get in the habit of looking for them, it’s not hard at all. We’ll get used to seeing them, and then make the leap that Fourier did.

时钟图中的二手波（图 1.1）实际上描绘了一种空间波。它代表了一个沿着时间箭头前进的时间波，但我实际上画的是一个左右延伸的空间波。傅立叶的伟大思想涵盖了空间和时间。波的种类取决于物理过程。如果它通过时间传播，如声音、光或二手波，那么这些波就是时间波，频率就是每秒的周期数。图片_二手波的频率是空间波，它的频率以每英寸的周期数表示，如图所示，大约每 3 英寸一个周期。如果我要以相同的比例绘制分针波，它的频率将减少 60 倍，或大约每 180 英寸一个周期。时针空间波大约每 2,160 英寸有一个周期，比分针波少十二倍。

The second-hand wave in the clock diagram (figure 1.1) actually depicts a space wave. It represents a time wave proceeding along time’s arrow, but what I’ve actually drawn is a space wave extending left and right. Fourier’s great idea covers both space and time. The kind of wave depends on the physical process. If it proceeds through time, like sound, light, or the second-hand wave, then the waves are time waves and the frequencies are cycles per second. The picture of the second-hand wave is a space wave, and its frequency is expressed in cycles per inch, roughly one cycle per 3 inches as drawn. If I were to draw a picture of the minute-hand wave at the same scale, its frequency would be sixty times less, or roughly one cycle per 180 inches. The hour-hand space wave would have roughly one cycle per 2,160 inches, twelve times less than the minute-hand wave.

我称之为视觉波的波不要与光波混淆。光波是一种激发我们眼睛的视杆和视锥细胞的机制，以便我们可以看到。它是一种以极高频率随时间变化的波，例如每秒 500 万亿个周期。所以光波是观看的手段，但视觉波是我们所看到的。光波随时间变化，视觉波随空间变化。

The waves I’m calling vision waves are not to be confused with light waves. A light wave is a mechanism for exciting the rods and cones of our eyes so that we can see. It’s a wave that varies in time at extremely high frequencies, such as 500 trillion cycles per second. So light waves are the means of seeing, but vision waves are what we see. Light waves vary in time, vision waves in space.

视觉空间波的证据无处不在。看看你正在阅读的页面。字母，包括作为字符的空格，似乎或多或少均匀地放置在页面的水平尺寸上。在frequencyspeak中，它们在页面上以或多或少的恒定频率出现，并且文本行在页面下方以绝对恒定的频率出现。您可以将文本视为波峰，将线条之间的空间视为某种“波浪”的波谷。书架上的书籍在水平方向上以相当恒定的速度重复。货架本身以相当可预测的频率垂直重复。这些都不是傅里叶的美丽平滑波，但它们都暗示了视野中空间波的概念。

Evidence of visual space waves is everywhere. Just look at the page you’re reading. The letters, including the spaces as characters, appear more-or-less uniformly placed across the horizontal dimension of the page. In frequencyspeak, they appear at a more-or-less constant frequency across the page, and the lines of text appear at a definitely constant frequency down the page. You can think of the text as the crests and the space between the lines as the troughs of some kind of “wave.” Books on a bookshelf repeat at a fairly constant rate horizontally. And the shelves themselves repeat at a fairly predictable frequency vertically. None of these is a beautiful smooth wave à la Fourier, but they all hint at the notion of spatial waves in the visual field.

事实上，如果你能在你的视觉世界中感知到某个频率的“波”，那么傅里叶版本的它就会以那个频率的波为特征。我刚刚测量了我正在阅读的书中一行文本的底部与下一行底部之间的距离为四分之一英寸。所以那个页面的音乐版本，如果你想它是一张图片，一定有一个频率相同的傅立叶空间波，每英寸四个周期。

In fact, if you can perceive a “wave” of some frequency in your visual world then the Fourier version of it will feature a wave of that frequency. I just measured the distance between the bottom of one line of text in a book I’m reading to the bottom of the next line to be a quarter inch. So the musical version of that page, which is a picture if you think about it, must have a Fourier space wave of that frequency, four cycles per inch.

屋顶梁以恒定的节拍“涟漪”穿过天花板，硬木地板的木板在一维中以熟悉的频率重复。瓷砖地板有重复的瓷砖在两个维度的平铺频率上，平铺屋顶也是如此。Alhambra（图 1.5）将瓷砖——地板、墙壁、天花板——带到了令人惊叹的愉悦高度。

Roof beams “ripple” across the ceiling at a constant beat, and a hardwood floor has boards repeating at a familiar frequency in one dimension. A tiled floor has tiles repeating at the tiling frequency in both dimensions, and a tiled roof does too. The Alhambra (figure 1.5) takes tiling—of floors, walls, ceilings—to stunning heights of delight.

妙语是，您可以仅用傅立叶波来表示所有这些空间节奏。所有视觉图案，无论多么明显不规则，都可以描述为傅立叶美丽规则波的组合、总和。

The punch line is that you can represent all these spatial rhythms with Fourier’s waves alone. All visual patterns, no matter how apparently irregular, can be described as a combination, a sum, of Fourier’s beautiful regular waves.

但是自然界呢？那里也是如此，尽管不太明显——除了海浪。草坪或草地上的草叶以特征频率重复。树的叶子也是如此，尽管叶子之间的粗略间距取决于树的种类。森林中树木的间距具有特征频率，也取决于物种。山脉以某种频率达到顶峰。大自然母亲的频率远不如我们的那么均匀。她的重复模式更具随机性。然而，山峰不会出现在例如每英寸两个山峰或每百万英里两个山峰处。有一个频率范围，这或多或少是我们对山脉的期望，更像是每十英里有两个山峰。再一次，所有这些视觉节奏都符合傅立叶的节奏。

But what about the natural world? The same is true there, although less obvious—except for ocean waves. The blades of grass in a lawn or meadow repeat at a characteristic frequency. The leaves of a tree do too, although the rough spacing between leaves depends on the species of tree. The spacing of trees in a forest has a characteristic frequency, also dependent on species. Mountain ranges peak in a sort of frequency. Mother Nature’s frequencies are much less uniform than ours. She has more randomness in her repeating patterns. Yet mountain peaks don’t occur at, say, two crests per inch or at two crests per million miles. There is a range of frequencies, which is more-or-less what we expect of mountain ranges, more like two peaks per ten miles. Again, all these visual rhythms beat to Fourier’s tune.

图 1.6 的目的是向您展示如何在世界上寻找空间频率——作为查看它的另一种方式。这是我多肉花园的照片加利福尼亚州伯克利。到处都有空间频率。一旦你开始在这里或任何地方看到它们，那么实现傅立叶所做出的飞跃并不难——波和它们的各种频率是你完全描述视觉场景所需要的。

The purpose of figure 1.6 is to show you how to look for spatial frequencies in the world—as an alternative way to view it. It’s a picture of my succulent garden in Berkeley, California. There are spatial frequencies everywhere in it. Once you begin to see them, here or anywhere, then it isn’t so difficult to make the leap that Fourier made—that waves and their various frequencies are all you need to completely describe a visual scene.

考虑一下陶罐，这是这个场景的一些人造成分。观察它们在松散的网格上的情况。它们的间距不规则，但它们确实有一个松散规则的重复模式——比如每隔一英寸左右就有一个锅。在这部分图片的傅里叶等效项中，有一组与底池放置频率略有偏离的波浪，以解释与严格规律性的轻微偏离。

Consider the terracotta pots, some of the manmade constituents of this scene. Observe how they are on a loose grid. They aren’t regularly spaced exactly, but they do have a repeat pattern that’s loosely regular—say a pot every one inch or so. In the Fourier equivalent of this part of the picture there is a group of waves with slight departures from that frequency of pot placement to account for the slight departures from strict regularity.

花盆上的图案也很规则。只考虑一条穿过图片的线，比如一条与罐子对齐的线，由下方的一对箭头表示，线 a。换句话说，一次只将注意力集中在一条线上的规则频率上——就好像沿着那条线的光强度是一段音乐中的声音幅度一样。你能想象你花园的光强度会说“是”吗？

The patterning on the pots is regular too. Consider only a single line through the picture, say a line aligned with the pots and indicated by the lower pair of arrows, line a. In other words, constrain your attention to the regular frequencies along one line at a time—as if the light intensities along that line were the sound amplitudes in a passage of music. Can you imagine a yes spoken by your garden’s light intensities?

从a线上的左边开始，详细地，那里是锅上螺旋凹槽的中频，然后是锅之间的高频（但幅度很小）污垢，然后是第二个锅上的凹槽的中频，然后是定义最大电位器曲线的波的低频，然后是其他两个电位器的两个稍高的频率，依此类推。

Starting from the left on line a, in detail, there is the medium frequency of the spiraling grooves on the pot there, then the high frequency (but low amplitude) dirt between the pots, then a medium frequency of the grooves on the second pot, then the low frequency of the wave defining the curve of the largest pot, then two slightly higher frequencies for the two other pots, and so forth.

植物也有频率。不同的仙人掌和多肉植物具有不同的分枝或叶子放置频率。要看到这一点，请将您的注意力集中在沿线 b 的位置处几乎规则的频率上。这条线穿过（图像）两个星爆状结构的仙人掌。沿着这条线，每个物种的“叶子”以几乎规律的频率出现。（仙人掌的刺实际上是它的叶子。“叶子”是它的树枝。）

The plants too have frequencies. The different cactus and succulent species have various frequencies of branch or leaf placement. To see this, constrain your attention to the almost regular frequencies at places along the line b. The line cuts through (the image of) two cacti of starburst-like structure. Along that line the “leaves” of each species occur at an almost regular frequency. (The cactus spines are actually its leaves. The “leaves” are its branches.)

下一条线，c 线，与绣球花的大球相交，每个球由数百个小花组成。小花以比任何其他尚未讨论的植物更高的频率与线相交。下一行（d 行）与绣球花的叶子相交。它们沿着这条线出现的频率低于小花沿着c线出现的频率。也就是说，同一个空间里大约有两个叶宽，相当于有几十个小花宽。小花比叶子更密集，或者，在频率峰值中，它们以更高的空间频率出现。

The next line up, line c, intersects the large balls of hydrangea flowers, each composed of hundreds of tiny florets. The florets intersect the line at a higher frequency than any other plant yet discussed. The next line up (line d) intersects the leaves of the hydrangea. They occur along that line at a lower frequency than the florets do along line c. That’s to say, there are about two leaf widths in the same space as there are dozens of floret widths. The florets are denser than the leaves or, stated in frequencyspeak, they occur at a higher spatial frequency.

穿过图片的四行显示为彼此平行，但傅立叶并不要求这样做。您可以通过在图片中以其他角度（甚至垂直于所示角度）寻找重复的图案来锻炼您的直觉。例如，您可能会看到大扁平椭圆上的凸起的规则结构照片中央附近的仙人掌仙人掌的“叶子”。那些颠簸是它的刺所在的地方。您可以通过二维中的凸块绘制两个或多或少相互垂直的波。这或多或少是二维傅里叶思想的作用。

The four lines through the picture are shown parallel to one another, but Fourier doesn’t require that. You might exercise your intuition by looking for repeating patterns in lines through the picture at other angles, even perpendicular to the ones shown. For example, you can probably see the regular structure of the bumps on the large flat oval “leaf” of the prickly pear cactus near the center of the photo. Those bumps are where its spines are located. You could draw two waves more or less perpendicular to one another through the bumps in the two dimensions. That’s more or less what the two-dimensional Fourier idea does.

回到人造部分：砖天井在宽度和宽度上有重复频率。背景中的栅栏（很难看到）有它的板重复频率，在它上面编织的格子也是如此。

Back to the manmade parts: The brick patio has repeat frequencies in width and breadth. The fence in the background (difficult to see) has its frequency of board repeats, and so does the trellis weaving above it.

我们可以继续对照片的每个部分进行这种分析。由于粗糙，穿过树皮的线会产生具有高频细节的波。盆中的豌豆砾石以非常高的空间频率出现。等等。花园是空间音乐的交响曲。

We could continue this analysis for every part of the photograph. A line through the tree bark would produce a wave with high frequency detail because of the roughness. The pea gravel in the pots occurs at very high spatial frequency. And so forth. Gardens are symphonies of spatial music.

傅立叶的遗产是我们所看到的一切——呈现给我们视网膜的视觉世界，无论它是否看起来有重复的模式——都是空间音乐的交响乐。它可以用所有频率和幅度的二维空间波来表示。都是音乐。它就像音乐一样工作，但对我们的眼睛来说是二维的，而不是对我们的耳朵来说的。我们需要这种关于可见世界波动性质的直觉，以便在下一章解释像素。

Fourier’s legacy is that everything we see—the visual world presented to our retinas, whether it appears to have repeated patterns or not—is a symphony of spatial music. It can be represented with two-dimensional space waves of all frequencies and amplitudes. It’s all music. It works just like music but in two dimensions for our eyes instead of one for our ears. We need this intuition about the wave nature of the visible world in order to explain pixels in the next chapter.

罗塞塔石碑

Rosetta Stone

让-弗朗索瓦·商博良是格勒诺布尔的公民，而傅立叶则应波拿巴的“要求”在那里担任省长。傅立叶向年轻的商博良介绍了罗塞塔石碑（图 1.7）以及占据其前三分之一的古埃及象形文字的奥秘。在接下来的二十年里，商博良使用石头底部三分之一处的古希腊对应物作为重要来源，破译了象形文字。

Jean-François Champollion was a citizen of Grenoble while Fourier was prefect there, at Bonaparte’s “request.” Fourier introduced young Champollion to the Rosetta Stone (figure 1.7) and the mystery of the ancient Egyptian hieroglyphs that occupy its top third. Over the next two decades Champollion deciphered the hieroglyphs using the ancient Greek equivalent in the bottom third of the stone as an important source.

傅立叶与第一领事的特殊关系使这成为可能。商博良多次被征召入伍，但每次傅立叶都成功地直接向当时的首席埃及亲信者波拿巴提出了豁免这个年轻人的请求。如此自由地追求他的激情，商博良破解了罗塞塔石碑的密码并建立了埃及学领域。9

Fourier’s special relationship with the First Consul made this possible. Champollion was drafted several times, but in every case Fourier successfully appealed directly to the chief Egyptophile at the time—Bonaparte—to exempt the young man. Thus free to pursue his passion, Champollion cracked the Rosetta Stone’s code and established the field of Egyptology.9

傅立叶的伟大想法——世界就是音乐，一切都是波——是科学的罗塞塔石碑，而傅立叶是它的商博良。今天，所有同类的科学家、工程师和技术人员都在使用傅里叶频率语言。它是声音、图像、视频等等的通用语言——包括许多最常见的物理过程，尤其是媒体技术。傅立叶展示了如何在他的频率语言和普通语言之间来回翻译，比如跨越空间的颜色或跨越时间的声音振幅。

Fourier’s great idea—the world is music, it’s all waves—is the Rosetta Stone of science, and Fourier was its Champollion. Scientists, engineers, and technologists of all ilk speak Fourier’s frequency language today. It’s the lingua franca of sound, image, video, and on and on—including many of the most common physical processes, and particularly the media technologies. Fourier showed how to translate back and forth between his frequency language and the ordinary languages of, say, colors across space or sound amplitudes through time.

您可能在随意的谈话中听到有人说“我没有足够的带宽”。如果您是技术人员，您将立即知道这意味着什么。如果你不是，你可能已经推断出它的意思是“容量”，但你不知道为什么。这是一个傅立叶频率词，这就是原因。

You’ve perhaps heard someone say, “I haven’t got enough bandwidth,” in casual conversation. If you’re technical, you’ll immediately know what it means. If you’re not, you’ve probably deduced that it means something like “capacity,” but you don’t have a clue why. It’s a Fourier frequency word, that’s why.

从技术上讲，带宽是衡量通信信道容量的指标。例如，年轻的人耳可以听到每秒 20,000 次循环到每秒 20 次循环的声音。所以它的带宽是两者之差，即对耳朵有意义的频带宽度。在随意的演讲中，带宽被用来比喻你处理信息的能力。

Bandwidth is, technically, a measure of the capacity of a communications channel. The young human ear, for example, can hear sounds of 20,000 cycles per second down to 20 cycles per second. So its bandwidth is the difference of the two, the width of the band of frequencies that are meaningful to the ear. In casual speech bandwidth is used metaphorically to mean something like your capacity to process information.

像这样的频率说话是两种文化的分隔符之一。它对艺术来说是希腊语，所以通过罗塞塔石碑的比喻，频率说话（傅立叶的波浪语言）必须是象形文字。现在让我们看看傅立叶的象形文字——他的二维形式的波——如何代表图片。

Frequencyspeak like this is one of the separators of the two cultures. It’s Greek to the arts, so by the Rosetta Stone metaphor, frequencyspeak (Fourier’s wave language) must be the hieroglyphs. Now let’s see how Fourier’s hieroglyphs—his waves in two-dimensional form—can represent pictures.

波纹和沟槽

Corrugations and Furrows

究竟什么是二维波？到目前为止，我们只看到了一维波，比如用于声音的波。傅立叶应用于视觉世界的想法需要他的波的二维版本。

What exactly is a two-dimensional wave? We’ve only looked so far at one-dimensional waves, like those used for sound. Fourier’s idea applied to the visual world needs a two-dimensional version of his wave.

一维波是一个展开的圆。二维波是一个展开的圆柱体。要在二维中可视化波浪，请想象时钟的二手波浪垂直于页面挤出。你得到的是一个波纹状的表面，就像一个有沟的田野。波纹金属建筑由二维空间波片组成。一些门廊屋顶由波纹塑料板制成（图 1.8）。某些褶边薯片是波纹状的。我们称之为瓦楞纸板的平面之间的波浪层是另一种空间波。地中海风格的红瓦屋顶是另一个。

A one-dimensional wave is an unwound circle. A two-dimensional wave is an unwound cylinder. To visualize a wave in two dimensions, imagine the clock’s second-hand wave extruded perpendicular to the page. What you get is a corrugated surface, like a furrowed field. A corrugated metal building is composed of sheets of two-dimensional space waves. Some porch roofs are made from corrugated plastic panels (figure 1.8). Certain ruffled potato chips are corrugated. The wavy layer between the flat surfaces of what we call corrugated cardboard is another space wave. A Mediterranean-style red tiled roof is another.

因此，波纹是二维空间波。查看它的边缘（横截面）以查看一维波。事实上，它的任何直截面也是一维波。

So, a corrugation is a two-dimensional space wave. Look at its edge—a cross-section—to see a one-dimensional wave. In fact, any straight cross-section of it is also a one-dimensional wave.

傅立叶的大思想是，所有的视觉世界都可以表示为所有频率和幅度的波纹空间波的总和。为了处理第二维，唯一增加的扭曲是波可以旋转到任何方向。犁沟可以向南北向，或东向西，或东北向西南，或以任何其他角度延伸，这对自然界尤其重要。空间波的“材料”可能是镀锌铁，或者，正如图中所示，塑料。工程师们使用傅立叶的想法来描述现实和人造世界中的所有复杂模式——由世界构成的所有材料构成。

Fourier’s big idea is that all of the visual world can be represented as a sum of—only—corrugated space waves of all frequencies and amplitudes. The only added twist, to handle the second dimension, is that the waves can be rotated to any degree of orientation. The furrows can run north and south, or east and west, or northeast by southwest, or at any other angle, especially important for the natural world. The “stuff” of the space waves might be galvanized iron or, as the figure here suggests, plastic. Engineers use Fourier’s idea to describe all the complex patterns in the real and manmade world—made of all the stuff the world is made of.

但是要理解像素，我们需要考虑某人看待那个世界的观点。世界可能是由铁、塑料和土豆构成的，但我们看到的是一种色彩亮度的图案——一个由明暗色调和形状组成的领域。我们所看到的世界的傅里叶波的东西是光的强度，因为它在视野中变化。

But to understand the pixel we need to consider the point of view of someone looking at that world. The world may be made of iron and plastic and potatoes, but what we see is a pattern of colored brightnesses—a field of light and dark hues and shapes. The stuff of Fourier’s waves for the world we see is the intensity of light as it varies across the visual field.

我们怎么能想象这样一个领域呢？解释呈现给眼睛的世界的一种明显方法是描述视野中每个点的光强度。但这意味着视野中的每个点。视野是一个连续的东西——每一个可能的点都被光着色。这些点之间没有中断。无论我们选择的两个点有多接近，中间总会有另一个点。有无数个点，需要指定很多点。

How might we even imagine such a field? One obvious way to explain the world that’s presented to the eye would be to describe the light intensity at every point of the visual field. But this means every point in the visual field. A visual field is a continuous thing—every possible point is colored with light. There are no breaks between these points. No matter how close together the two points we choose, there’s always another one in between. There’s an infinite number of points and that’s a lot of points to specify.

傅里叶为我们提供了另一种方式：将视觉世界视为空间波的总和，这些空间波加起来与逐点描述相同。例如，图 1.9 顶行中的图片可以通过给出每个点的灰度值以显而易见的方式描述，或者可以通过仅指定一个波等效地描述为傅里叶方式。“加起来”到这张图片中的一个空间波的频率约为每英寸两个周期，其亮度水平在明暗之间摆动。只有少数几个值指定了整个图片——一个频率和一个最大幅度——而不是大量的灰度，一个对应图片中的每个点。

Fourier offers us an alternative way: to think of the visual world as a sum of spatial waves that add up to be the same as the point-by-point description. The picture in the top row of figure 1.9, for instance, could be described in the obvious way by giving its gray value at every point, or it could equivalently be described the Fourier way by specifying just one wave. The one space wave that “adds up” to this picture has a frequency of, say, about two cycles per inch, and its brightnesses wiggle between dark and light horizontally. Just a handful of values specifies this entire picture—one frequency and one maximum amplitude—instead of an extremely large number of grays, one for each point in the picture.

顺便说一句，最上面的图片显示了从上面看到的傅里叶波纹之一（部分）。因此，毫不奇怪，那个波浪就是这张图像的完整描述。请注意，您可以看到构成视野的波的振动，就像您可以听到构成声音的波的振动一样。该图中波的“成分”是光强度（或亮度）——被认为是灰色阴影。波浪的高部分是浅灰色，低部分是较暗的。（该图是图 1.2 中波浪线的二维对应物。）

By the way, that top-row picture shows (part of) one of Fourier’s corrugated waves seen from above. So, not too surprisingly, that very wave is the complete description à la Fourier of this image. Notice that you can see the vibrations of the waves that make up the visual field, just as you can hear the vibrations of those that make up sound. The “stuff” of the wave in this figure is light intensity (or brightness)—perceived as a shade of gray. High parts of the wave are lighter grays, and low parts are darker. (This figure is a two-dimensional counterpart of the wavy lines in figure 1.2.)

幅度对应于视觉场景的亮度，就像它对应于声音的响度一样。频率对应于场景中细节的数量，人们在谈论高分辨率图片或高清视频时所指的那种东西。高振幅意味着明亮，高频意味着细节。

Amplitude corresponds to the brightness of a visual scene, just as it corresponds to loudness in the case of sound. Frequency corresponds to the amount of detail in a scene, the sort of thing people refer to when they talk about a high-resolution picture or a high-definition video. High amplitude means bright, and high frequency implies detail.

当颜色可用时，颜色信息也在那里。我们的眼睛有三种颜色感受器——视锥细胞——红、绿、蓝光强度各有一种。如果一个手机或电视屏幕上的点会显示红、绿、蓝光强度，然后我们的大脑会从我们的视锥细胞中接收这三种信号的相对强度，并从中感知颜色。来自电子显示器的发射光混合颜色的方式与来自油画（比方说）或彩色墨水印刷页面的反射光不同。为了使发射的光呈现黄色，我们必须打开红色和绿色部分并关闭蓝色部分。但是，为了让白色页面反射的光呈现黄色，我们必须阻挡白光的蓝色部分，只让红色和绿色通过。无论采用何种显示技术，在视觉上都是一样的。我几乎总是使用发射（非反射）光术语。

Color information is there too, when color is available. Our eyes have three types of color receptors—the cones—one type each for red, green, and blue light intensities. If a point on a cellphone or television screen displays red, green, and blue light intensities, then our brain takes in the relative strengths of these three signals from our cones and perceives a color from them. Emitted light from electronic displays mixes colors in a different way than does reflected light from an oil painting, let’s say, or a printed page of colored inks. For emitted light to appear yellow, we must turn on the red and green parts and turn off the blue part. But for light reflected off a white page to appear yellow, we must block the blue part of the white light and let only the red and green through. Regardless of the display technology, it all amounts to the same thing at the eye. I’ll nearly always use emitted (not reflected) light terminology.

因此，将图 1.9 顶行中的灰度波想象成一种颜色波——比如说黄色波。所显示的波峰是介于中灰和全白之间的灰色，而我们的彩色波峰是介于中黄和全黄之间的黄色。您可以将彩色波想象成三个频率相同但幅度不同的波，每个用于我们的红色、绿色和蓝色受体。所以我们的黄色波由三个波组成，一个类似于上面显示的红色受体的灰度波，另一个与绿色受体类似（相同幅度），第三个与蓝色受体类似但幅度为零。红色和绿色相等，没有蓝色，我们的大脑看到黄色波。

So imagine the grayscale wave in the top row of figure 1.9 as a wave of color instead—say a yellow wave. Where the crest of the wave shown is a gray somewhere between mid-gray and full white, the crest of our colored wave is a yellow somewhere between mid-yellow and full yellow. You can think of a colored wave as three waves at the same frequency but of different amplitudes, one each for our red, green, and blue receptors. So our yellow wave consists of three waves, one like the grayscale wave shown above for the red receptor, another just like it (at the same amplitude) for the green receptor, and a third like it but with zero amplitude for the blue receptor. With equal parts red and green and no blue, our brain sees a yellow wave.

图 1.9 中下两行的图片显示了额外的二维波浪，以使您熟悉它们的外观。就像一维的情况一样，三个波处于三个不同的频率和三个不同的幅度。对于所示的波，最高频率波（底行）具有最低振幅（中灰色），而最低频率波（顶行）具有最高振幅（接近白色）。但事实并非如此。波可以具有任何幅度的任何频率。

The pictures in the middle and bottom rows of figure 1.9 show additional two-dimensional waves to familiarize you with what they look like. Just as for the one-dimensional case, the three waves are at three different frequencies with three different amplitudes. For the waves shown, the highest frequency wave (bottom row) has the lowest amplitude (mid-gray), and the lowest frequency wave (top row) has the highest amplitude (near white). But that doesn’t have to be the case. A wave can have any frequency at any amplitude.

这是从这些数字中得出的直觉：图片中的精细细节必须来自其傅里叶描述中的高频波。它们是唯一变化得足够快的波浪。计算机图形学高手会说诸如“那个场景中有很多高频”之类的东西，这意味着——当然没有解释——它有很多微小的细节和锐利的边缘。

Here’s the intuition to take away from these figures: the fine detail in a picture must come from the high-frequency waves in its Fourier description. They are the only waves that change fast enough. Computer graphics whizzes say things in frequencyspeak like, “There are lots of high frequencies in that scene,” meaning—without explanation, of course—that it has many tiny details and sharp edges.

后面关于边缘的评论指的是你应该知道的一个方便的事实。这当然不直观。这是傅里叶数学的直接结果：在锐边的突然转变需要傅里叶世界中非常高的频率；一个非常突然的变化需要非常高的频率。

The latter remark about edges refers to a handy fact you should know. It’s certainly not intuitive. It’s a direct consequence of Fourier’s math: the sudden transition at a sharp edge requires very high frequencies in Fourier’s world; a really sudden change requires a really high frequency.

这是我对傅里叶妙语的再次表述：

Here’s my formulation of Fourier’s punch line again:

任何视野——让我们称之为图片或图案——都是优雅起伏的波浪的总和，并且只是这些波浪的总和，所有这些都是完美圆柱体的展开。

Any visual field—let’s call it a picture or a pattern—is a sum of, and only of, gracefully undulating waves such as those shown, all unfurlings of perfect cylinders.

作为波浪的视觉世界并不比声音更神秘——或者我们可以说它同样非常神秘。葛底斯堡演说可以用林肯演讲时每一刻的声压强度来描述，也可以用不同频率和振幅的声波叠加在一起来描述。傅立叶告诉我们两种描述是等价的。

The visual world as waves is no more mysterious than sound—or let’s say it’s just as wonderfully mysterious. The Gettysburg Address can be described by the intensity of sound pressure at each moment during Lincoln’s speech or it can be described as the sum of the sound waves of various frequencies and amplitudes that add up to the same thing. Fourier taught us that both descriptions are equivalent.

科学家和工程师喜欢傅立叶的版本，因为通过使用非显而易见的频率描述，他们可以解决现实世界中无法通过明显的、等效的逐点描述解决的问题。

Scientists and engineers love Fourier’s version because, by using the nonobvious frequency description, they can solve problems in the real world that resist solution in the obvious, equivalent point-by-point description.

我们不是在谈论一千个光点，而是在频率上谈论叠加到这些点的光强度波的频率和幅度。傅立叶的变革性教导是这两种描述是等价的。逐点视野是希腊文。傅立叶波音乐等价物是象形文字。对于极客来说，象形文字比希腊文更容易阅读。罗塞塔石碑是傅立叶证明两者相同的证据。

Instead of talking about a thousand points of light, we talk in frequencyspeak about the frequencies and amplitudes of the waves of light intensity that add up to the points. Fourier’s transformative teaching is that these two descriptions are equivalent. The point-by-point visual field is the Greek text. The Fourier wave musical equivalent is the hieroglyphs. For a geek, hieroglyphs are easier to read than Greek. The Rosetta Stone is Fourier’s proof that the two are the same.

傅立叶的批评者不相信他是对的，但数学表明这是对的。这就是这个想法的魔力——以及数学令人敬畏的、打破直觉的力量。把不同频率的波加起来真的会给你一张图片。. . 任何事物！例如，关于我多汁的花园，或者你正在阅读的页面，或者你的孩子。这是傅立叶伟大而宏大的想法。

Fourier’s critics didn’t believe he was right, but the math said it was true. That’s the magic of the idea—and the awesome, intuition-breaking power of math. Adding up waves of different frequencies really will give you a picture of . . . anything! Of my succulent garden, or the page you’re reading, or of your child, for instance. That’s Fourier’s great and very large idea.

需要热量

A Need for Heat

傅立叶在恐怖袭击中幸存下来，他甚至在法国统治——如果只是作为省长——多亏了波拿巴。但是，当国王重新掌权时，无论是他早期的反贵族立场还是他与皇帝的关系，都不是方便的凭据——两次。尽管如此，傅立叶还是在这场摄政和帝国之舞中的幸存者，就像他在恐怖中一样。他以相当高的政治技巧成功地在伊泽尔省担任了近十三年的省长，只有最后一年是冒险的。

Fourier survived the Terror, and he even came to rule in France—if only as a provincial governor—thanks to Bonaparte. But neither his early anti-aristocratic stance nor his association with the emperor were handy credentials when the king returned to power—twice. Nevertheless, Fourier was a survivor in this dance of regency and empire, as he had been in the Terror. He managed, with considerable political skill, to remain prefect of the Isère department for almost thirteen years, and only the last year was dicey.

在 1802 年 4 月至 1814 年 4 月的 12 年中，傅立叶作为一名出色的格勒诺布尔总督充分利用了他的流放生涯。他与40个公社谈判达成了一项协议，以排干伊泽尔 (Isère) 布尔干 (Bourgoin) 的大沼泽地，这是一场巨大的政变，因为之前的所有谈判尝试都失败了。他穿过一条从格勒诺布尔到都灵的新路。他为图书馆购买书籍，支持他所在部门的年轻人——最著名的是商博良——并参与了《埃及的描述》 ，最终于 1810 年出版。在此期间，傅立叶以某种方式找到了发展他的波动理论的时间。他不是通过听莫扎特或观察阿尔罕布拉宫到达那里的，而是通过考虑热量波的条款。也许他的théorie de la chaleur（他的热理论）——以及他伟大的和谐思想——是他重新进入巴黎的总体计划。

For 12 years, April 1802 to April 1814, Fourier made the best of his exile as an accomplished governor in Grenoble. He negotiated an agreement with 40 communes to drain the massive swamp of Bourgoin in Isère, quite a coup since all previous attempts at negotiation had failed. He pushed through a new road from Grenoble to Turin. He purchased books for the library, championed young men of his department—most notably Champollion—and worked on Déscription de l’Egypte, finally published in 1810. During this interval Fourier somehow found time to develop his wave theory. He didn’t get there from listening to Mozart or observing the Alhambra but by thinking of heat in terms of waves. Perhaps his théorie de la chaleur (his theory of heat)—and with it his great harmonic idea—was his master plan for readmission to Paris.

傅立叶本人谈到他对热的非凡兴趣时说：“在我看来，地球温度问题一直是宇宙学研究的最重要对象之一，而我在建立热的数学理论时主要考虑了这个主题。” 他正在寻找一个和牛顿一样大的想法。

Fourier himself said about his extraordinary interest in heat, “The question of terrestrial temperatures has always appeared to me to be one of the greatest objects of cosmological studies, and I have had this subject principally in view in establishing the mathematical theory of heat.” He was looking for an idea as big as Newton’s.

Victor Cousin 是一个认识傅立叶的人，他对他的动机有不同的看法。他写道，当傅立叶从埃及回到格勒诺布尔时，即使在最热的天气里，他也从不出门，没有一件大衣和另一件备用。他已经产生了对热量的身体需求，这听起来充其量是古怪的，但这相当于一种疾病。

Victor Cousin, a man who knew Fourier, had a different take on his motivation. He wrote that when Fourier came back to Grenoble from Egypt, he never went outside, even in the hottest weather, without an overcoat and another in reserve. He’d developed a physical need for heat that sounds eccentric at best, but which amounted to a disease.

法国人对他有一个我们在英语中缺乏的词：傅立叶是frileux。他感到非常寒冷。他对寒冷非常敏感，也许很痛苦。这个词还带有他无法个人温暖的内涵。人们可能认为傅立叶的冷酷程度不止一种。

The French have a word for him that we lack in English: Fourier was frileux. He felt the cold terribly. He was very sensitive to cold, perhaps painfully so. The word also carries the connotation that he was incapable of personal warmth. People may have thought that Fourier was chilly in more ways than one.

没有人知道傅立叶是什么时候开始研究热力的，但他在格勒诺布尔时就全神贯注于这个理论，就在埃及之后几年。可能在 1804 年或 1805 年完成的有缺陷的草稿版本包含他的波浪的第一次已知外观。他在 1807 年末的一本名为“固体中的热传播理论”的“回忆录”中向学术公众展示了该手稿的一个大大改进的版本，现在被认为是确立他的想法的论文。在编写这两个版本之间的时间里，他已经熟悉了热的实际物理学，并进行了实验来测试和验证他的数学结果。

Nobody knows when Fourier first started his work on heat, but he was engrossed in the theory while in Grenoble, just a couple years after Egypt. A flawed draft version probably completed in 1804 or 1805 contains the first known appearance of his waves. He presented an enormously improved version of that manuscript to the academic public in a late 1807 “memoir” called Theory of Propagation of Heat in Solids, now considered to be the paper that established his idea. During the time between writing the two versions, he had become familiar with the actual physics of heat and performed experiments to test and verify his mathematical results.

用傅立叶的话说，整个过程“有助于赋予该理论一种权威，在一个仍然晦涩难懂且显然存在许多不确定性的问题上，人们可能倾向于拒绝这种权威。” 这就是经典科学。使用理论进行假设。用实验来验证。这证明傅里叶既是数学家又是实验物理学家。他不怕弄脏自己的手。

In Fourier’s words, the whole process “contributed to give the theory an authority which one might have been inclined to refuse in a matter still obscure and apparently subject to so many uncertainties.” That’s classic science. Use theory to hypothesize. Use experiment to verify. It proves that Fourier was both a mathematician and an experimental physicist. He wasn’t afraid to get his hands dirty.

傅立叶意识到他可以将流经固体物体的复杂热量模式描述为他的波的总和。这意味着他可以预测热量将如何（以及多长时间）从大炮的接触点流到嘴巴。

Fourier realized that he could describe the complex pattern of heat flowing through a solid object as a sum of his waves. That meant that he could predict how (and how long) heat would flow from, say, the touchpoint of a cannon to its mouth.

因此，从长远来看，在 1803 年至 1807 年期间，他在完善热理论和进行支持性实验的同时，也在谈判沼泽排水合同、修路、指导部门中的年轻人，并为埃及出版物工作. 傅里叶从哪里获得带宽？

So, for perspective, between 1803 and 1807, while he was perfecting his theory of heat and performing the supporting experiments, he was also negotiating the swamp-draining contracts, building roads, mentoring young men in his department, and working on the Egypt publication. Where did Fourier get the bandwidth?

傅立叶驾驭了帝国和革命的政治。他利用自己对拿破仑（皇帝被称为波拿巴的称呼）的个人知识，根据一页纸的提案获得了大型公路项目的批准，拿破仑在两天内就批准了。但傅立叶在学术界更加激烈的政治中度过了一段艰难的时光，也许是因为流亡中的傅立叶与巴黎喧嚣的学术活动疏远了。

Fourier navigated the politics of empire and revolution. He used his personal knowledge of Napoleon—as Bonaparte the emperor was called—to get the big road project approved based on a one-page proposal, which Napoleon granted in two days. But Fourier had a harder time with the even fiercer politics of academia, perhaps because Fourier in exile was alienated from the hurly-burly of academic goings on in Paris.

在 1807 年“回忆录”的科学院官方读者中，有拉普拉斯，他是拿破仑的参议员任命者，他从傅立叶在巴黎综合理工学院的时候就认识了他。拉普拉斯对傅立叶伟大想法的数学运算感到不舒服。对它的第一次公开攻击来自拉普拉斯的门徒西蒙·丹尼斯·泊松，当傅立叶成为伊泽尔省长时，他在巴黎综合理工学院担任傅立叶的主席。长期而激烈的争论的结果是，这本回忆录从未出版过。最终，拉普拉斯开始支持傅立叶，但傅立叶的死敌泊松从来没有这样做过。

Among the official Academy of Science readers of the 1807 “memoir” was Laplace, Napoleon’s senatorial appointee, who had known Fourier since his days at the École polytechnique. Laplace was uncomfortable with the mathematics of Fourier’s great idea. The first public attack on it came from Laplace’s protégé, Simeon Denis Poisson, who assumed Fourier’s chair at the École polytechnique when Fourier became prefect of Isère. The outcome of the long and acrimonious controversy was that the memoir was never published. Eventually, Laplace came to support Fourier, but Poisson—Fourier’s implacable enemy—never did.

在后人看来，傅立叶在他 1807 年的回忆录中确立了这一理论——使 2007 年成为这个想法不为人知的 200 周年——但在它被真正接受之前，他必须经历另一个挑战。可能是因为长期的争论，学院宣布将在 1811 年颁发数学大奖，主题是 . . . 热在固体中的传播。厚脸皮！傅立叶以他的论文奖作为回应，这是 1807 年论文的扩展版本，保留了早期回忆录的主要贡献。10

To posterity’s mind, Fourier had established the theory in his memoir of 1807—making 2007 the idea’s uncelebrated bicentennial—but he had to pass through another gauntlet before it was truly accepted. Probably because of the prolonged controversy, the academy announced that it would award a grand prize in mathematics in 1811 on the subject of . . . the propagation of heat in solid bodies. Cheeky! Fourier responded with what is known as his Prize Essay, the extended version of the 1807 paper that kept the principal contributions of the earlier memoir in place.10

因此，让我们总结一下傅立叶在准备获奖论文时所做的事情：沼泽。道路。指导。描述 de l'Egypte。同样，他从哪里获得带宽？

So let’s summarize what Fourier did while he was preparing the Prize Essay: Swamps. Roads. Mentoring. Déscription de l’Egypte. Again, where did he get the bandwidth?

1807 年回忆录之外的获奖论文的延伸之一讲述了傅立叶的牛顿野心。他将他的热流理论应用于一个球体——一个行星大小的球体——因此是第一个研究现在被称为温室效应的陆地现象的人。穿过地球大气层的阳光加热地球表面，而大气层在附近保存了一些热量。这是一个不完美的比喻，但大气就像温室的玻璃，它可以让阳光进入，但不会让所有的热量散发出去。自然温室效应的热量使地球变暖，并使其成为我们所知的生命之地。尽管傅立叶才华横溢，但他无法预见目前威胁地球生命的非自然温室效应热量。11

One of the extensions of the Prize Essay beyond the 1807 memoir speaks to Fourier’s Newtonian ambitions. He applied his theory of heat flow to a sphere—a planet-sized one—and so was the first to study the terrestrial phenomenon now known as the greenhouse effect. Sunlight that passes through Earth’s atmosphere heats the Earth’s surface, and the atmosphere holds some of the heat nearby. It’s an imperfect metaphor but the atmosphere resembles the glass of a greenhouse, which allows sunlight in but doesn’t let all the heat out. The heat of the natural greenhouse-effect warms the Earth and makes it the place for life as we know it. Fourier, despite his brilliance, couldn’t have foreseen the unnatural greenhouse-effect heat that currently threatens life on Earth.11

尽管有学院派的恶作剧，傅立叶还是凭借他的论文奖获得了奖项。但同样，他没有得到阅读它的学院委员会的全力支持。特别是，拉普拉斯投了反对票。

Despite the academy shenanigans, Fourier won the prize with his Prize Essay. But again, he didn’t get the full support of the academy commission who read it. In particular, Laplace voted against it.

“作者得出这些方程式的方式并非没有困难，”委员会在其报告中说。“他整合它们的分析在一般性甚至严谨性方面仍有待改进。”

“The manner in which the author arrives at these equations is not exempt of difficulties,” the commission said in its report. “His analysis to integrate them still leaves something to be desired on the score of generality and even rigor.”

数学上缺乏严谨性是最大的侮辱。学院在出版有奖论文方面拖了后腿，就像它为回忆录所做的那样。12

Lacking rigor in math is the ultimate insult. The academy dragged its heels in publishing the Prize Essay as it had for the memoir.12

直到 1815 年拿破仑最后一次流放至圣赫勒拿，傅立叶返回巴黎后，他才成功发表了他的获奖论文。它仍然受到无情的泊松的猛烈抨击。但是傅立叶在泊松的著作中发现了一个基本错误，并且在他的竞争热理论中发现了一个错误的主张。傅立叶在给拉普拉斯的一封信中摧毁了两者。这是他的对手的致命一击，最终拉普拉斯站在了他的一边。

It wasn’t until after Napoleon’s final exile, to Saint Helena in 1815, and Fourier’s return to Paris, that he managed to get his Prize Essay published. It still came under withering fire from the implacable Poisson. But Fourier found an elementary error in Poisson’s writings and a false claim in his competing theory of heat. Fourier demolished both in a letter to Laplace. That was the coup de grâce to his opposition, finally bringing Laplace to his side.

与暴君共舞

Dancing with Tyrants

胜利的科学进步并没有与光荣的政治成功相匹敌。傅立叶继续与拿破仑共舞，这让他作为伊泽尔省长的第十三年也是最后一年感到不安。拿破仑于 1814 年 4 月退位，路易十八恢复为国王。这位前皇帝前往厄尔巴岛进行他的第一次流放，厄尔巴岛位于科西嘉岛以东约 30 英里处，位于巴黎东南部。他在去格勒诺布尔的路上自然会经过格勒诺布尔。在正常情况下，傅立叶也很自然地会向“给他”工作的人打招呼。知府必须迎接前皇帝。但时代几乎不正常。这注定是一次尴尬的相遇。

Triumphant scientific progress wasn’t matched by glorious political success. Fourier’s continuing dance with Napoleon upset his thirteenth and final year as prefect of Isère. Napoleon abdicated in April 1814, and Louis XVIII was restored as king. The ex-emperor headed for his first exile in Elba, an island about thirty miles straight east of Corsica and southeast of Paris. It was natural that he would pass through Grenoble on his way there. Under normal circumstances it would also be natural for Fourier to greet the man who had “given” him the job. The prefect would have to greet the ex-emperor. But the times were hardly normal. It promised to be an embarrassing encounter.

国王的复辟使傅立叶的地位暂时保持不变，但拿破仑会认为他向摄政的鞠躬是一张变脸吗？傅立叶不打算四处寻找。在幕后，他在拿破仑的旅行路线上绕道绕过格勒诺布尔，理由是对前皇帝的潜在伤害。有效。拿破仑回避了傅立叶，新的王室政府着手确认傅立叶为省长。国王的兄弟，即后来的查理十世，拜访了他，完成了交易。

The restoration of the king had left Fourier’s position provisionally intact, but would Napoleon consider his bow to regency a volte face? Fourier wasn’t about to hang around to find out. Behind the scenes he engineered a detour in Napoleon’s travel route to bypass Grenoble, citing potential harm to the ex-emperor. It worked. Napoleon sidestepped Fourier, and the new royal government proceeded to confirm Fourier as prefect. The king’s brother, who would become Charles X, paid him a visit, consummating the deal.

但这并不是与拿破仑共舞的结束。1815 年 3 月 2 日，傅立叶收到了来自邻近省长的这封可怕的信：

But that wasn’t the end of the dance with Napoleon. On March 2, 1815, Fourier received this frightening letter from a neighboring prefect:

我荣幸地通知您，波拿巴率领 1,700 人昨天在胡安湾登陆。. . [并且] 将由圣瓦利耶、迪涅和格勒诺布尔前往里昂。不管这个消息在你看来多么不同寻常，它都是真实的。13

I have the honor to inform you that Bonaparte at the head of 1,700 men disembarked yesterday at Gulf Juan . . . [and] is heading for Lyon by Saint-Vallier, Digne, and Grenoble. No matter how extraordinary this news may seem to you it is entirely true.13

拿破仑从厄尔巴岛回来了他最后的百日权力，再次取代了国王。他沿着连接厄尔巴岛和科西嘉岛到格勒诺布尔、欧塞尔和巴黎的那条决定性的东南西北走廊返回。傅立叶将不得不再次与拿破仑打交道，而此时他已经设法伪装成忠诚的保皇党人。

Napoleon returned from Elba for his final Hundred Days of power, displacing the king again. He returned along that fateful southeast-northwest corridor joining Elba and Corsica to Grenoble, Auxerre, and Paris. Fourier was going to have to deal with Napoleon again, just when he had managed to pass himself off as a loyal royalist.

他准备了府邸，供归来的皇帝居住。他在那里给他留下了一张漂亮的字条，给予他个人的欢迎。在其中，他清楚地表明了他对国王的新忠诚并解释了他的冲突。但是当拿破仑进入城市的前门时，他正在从格勒诺布尔的后门到里昂。

He prepared the prefectural residence for occupancy by the returning emperor. He left a nice note there for him, giving him a personal welcome. In it he stated clearly his new loyalty to the king and explained his conflict. But he was hightailing it out the back gate of Grenoble to Lyon as Napoleon entered the front gate of the city.

拿破仑非常不满，解雇了傅立叶。一个篡位者——傅立叶自己对拿破仑的评价——怎么可能解雇任何人？他不仅这样做了，而且傅立叶和格勒诺布尔的市民都认为他有权这样做，这是衡量这个人权力的标准。

Highly displeased, Napoleon fired Fourier. How could a usurper—Fourier’s own word for Napoleon—fire anybody? It’s a measure of the man’s power that he not only did so, but that Fourier and the citizens of Grenoble perceived he had the authority to do so.

顾问们通过向拿破仑展示傅立叶对Déscription de l'Egypte的赞誉“历史导论”来安抚拿破仑。他的怒火冷却下来，他要求见傅立叶。他们在布尔根相遇，傅里叶在那里排干了沼泽。很难理解这个删除的动态：数学家的软肋？老埃及同志？但是拿破仑不仅原谅了傅立叶，还让他成为了里昂的罗纳省长官！

Advisers mollified Napoleon by showing him Fourier’s laudatory “Historical Introduction” to Déscription de l’Egypte. His anger cooled, and he asked to see Fourier. They met in Bourgoin, where Fourier had drained the swamps. It’s difficult to understand the dynamics at this remove: A soft spot for mathematicians? Old Egyptian comrades? But Napoleon not only forgave Fourier, he made him prefect of Rhône, governed from Lyon!

罗讷河谷的任命发生在 1815 年 3 月 12 日，就在宣布拿破仑从厄尔巴岛返回的信之后十天。事情发生得非常快。拿破仑的滑铁卢是 6 月 18 日，路易十八的第二次复辟是 7 月 8 日。

The Rhône appointment occurred on March 12, 1815, just ten days after the letter announcing Napoleon’s return from Elba. Things were happening very fast. Napoleon’s Waterloo was June 18, and the second restoration of Louis XVIII was July 8.

拿破仑的第二次帝制只持续了著名的百日，而傅立叶的第二府更短，大约六十天。拿破仑作为皇帝的最后一项行为是在滑铁卢前两周授予傅立叶退休金，并在两周后生效。当然，这笔养老金从未兑现。因此，当拿破仑最终被流放到圣赫勒拿岛时，傅立叶既没有工作，也没有退休金，而且政治声誉也很差。

Napoleon’s second imperial effort lasted only the famous Hundred Days, but Fourier’s second prefecture was even shorter, about sixty days. One of Napoleon’s last acts as emperor was to grant Fourier a retirement pension just two weeks before Waterloo, to go into effect two weeks after it. Of course, that pension was never honored. So, when Napoleon was finally exiled to Saint Helena, Fourier was left without a job or pension and with a soiled political reputation.

但至少他的流放终于结束了。他可以回到巴黎并建立他的热音乐理论。

But at least his exile was finally over. He could return to Paris and establish his musical theory of heat.

约瑟夫和苏菲

Joseph and Sophie

傅立叶有生命吗？也许不在格勒诺布尔省，但肯定会回到首都？我们可以希望世界音乐概念之父有一部戏剧性的浪漫故事，就像大约十年前本杰明富兰克林与一家著名法国沙龙的寡妇的故事一样。

Did Fourier have a life? Perhaps not in provincial Grenoble but surely back in the capital? We could wish for the father of the world-as-music concept a dramatic romance, like Benjamin Franklin’s about a decade earlier with the widowed hôtesse of a renowned French salon.

也许又是那些奇怪的痛苦，但傅里叶没有赫尔维修斯夫人。事实上，可悲的是，似乎根本没有任何浪漫的兴趣。我们对亲密关系的唯一暗示——这肯定是柏拉图式的——是他与著名数学家玛丽-索菲·热尔曼的深厚友谊。她赢得了科学院奖，是第一位获得该奖项的女性，但她没有出席领奖，这让“一大群无疑被渴望看到一种新的精湛技艺的人所吸引”感到失望。但这还不足以让她获得参加会议的邀请。在傅立叶成为常任秘书七年后，她才收到参加学院降神会或会议的长期邀请。傅立叶做到了。

Perhaps it was those strange afflictions again, but there was no Madame Helvétius for Fourier. In fact, sadly, there seems never to have been any romantic interest at all. The only hint we have of an intimate relationship—and it was surely platonic—was his deep friendship with the noted mathematician Marie-Sophie Germain. She had won an Academy of Science prize, the first woman to do so, and had disappointed “a very large crowd that doubtlessly had been attracted by a desire to see a new kind of virtuosity,” by not showing up to receive the prize. But that hadn’t sufficed to get her an invitation to meetings. She only received a standing invitation to attend séances, or meetings, of the academy seven years later after Fourier became permanent secretary. Fourier made it happen.

从傅立叶写给玛丽-索菲的信件中，仅存一份。所有的都是正式的，除了一个，可能是在他晚年写的，因为字迹残缺，写给“Ch。S”（Chère Sophie）并签名“Jh”（约瑟夫）。但傅立叶给一位医生写了一封信，为一位身患疾病的亲爱但未透露姓名的朋友寻求他的秘密帮助：

There is one surviving cache of letters from Fourier to Marie-Sophie. All are formal except one, probably written in his last years, given the broken handwriting, addressed to “Ch. S” (Chère Sophie) and signed “Jh” (Joseph). But Fourier wrote a letter to a doctor seeking his confidential help for a dear but unnamed friend in medical trouble:

由于最稀有和最美丽的品质，她值得你所有的兴趣。对于温柔地爱着她的我自己，尽管这完全无法预料的事件可能会使我对她的感情化为乌有，但我非常感谢您为她和我所做的一切。14

She is worthy of all your interest by reason of the rarest and most beautiful qualities. For myself, who love her tenderly, in so much as this entirely unforeseen event may bring to nothing the feelings which I have had for her, I would be most deeply grateful for anything you could do for her and for me.14

我们不确定这个女人是谁，但杰曼确实死于乳腺癌。如果她真的是苏菲，他对她的感情会带来什么？15

We don’t know for sure who this woman was, but Germain did die of breast cancer. What something might the feelings he’d had for her, if indeed she was Sophie, have brought?15

成为不朽

Becoming Immortal

巴黎科学院最终让傅立叶成为成员（图 1.10）。拿破仑于 1821 年去世后不久，他的理论的完整版本最终以《热分析理论》的形式出版。荣誉开始累积。学院选举他为常任秘书，牛顿皇家学会于 1823 年 11 月任命让·巴蒂斯特·约瑟夫·傅立叶为外国会员。他在 21 岁时非常羡慕牛顿，但在 55 岁时，他终于在他生命的最后十年中踏上了永生之路自己。1826 年，他成为法兰西学院的官方常任理事，1889 年，古斯塔夫·埃菲尔将他的名字命名在这座著名的塔楼上。 16

The Academy of Science in Paris finally made Fourier a member (figure 1.10). Shortly after Napoleon’s death in 1821, the full version of his theory was finally published as the Analytical Theory of Heat. The accolades began to accumulate. The academy elected him permanent secretary, and Newton’s Royal Society made Jean Baptiste Joseph Fourier a Foreign Member in November 1823. He had so envied Newton at age 21, but at 55 he was at last, in his final decade, treading the road to immortality himself. He became an official immortel of the Académie française in 1826, and Gustave Eiffel placed his name on the famous tower in 1889.16

拿破仑的死也最终使养老金成为可能。起初，国王政府对傅立叶接受拿破仑在最后百日里担任罗讷省的任命并不高兴。在拿破仑流放圣赫勒拿岛后，他第一次尝试从君主制中获得养老金。他的请求被拒绝了。他在 1816 年、1818 年和 1821 年再次尝试，每次都被拒绝。只有在拿破仑死后，他的下一次尝试才成功。

Napoleon’s death also finally made a pension possible. At first, the king’s government wasn’t happy that Fourier had accepted Napoleon’s appointment to the prefecture of Rhône in the last Hundred Days. He first tried for a pension from the monarchy just after Napoleon’s exile to Saint Helena. His request was rejected. He tried again in 1816, 1818, and 1821, each time rejected. Only his next attempt, after Napoleon had died, was successful.

政治斗争已经结束，但与健康有关的斗争还没有结束。他从来没有摆脱某种身体上的瘟疫。作为一个上了年纪的人，他被呼吸问题所困扰，以至于他为自己制造了一个特殊的盒子，可以在他写作或说话时——或睡觉时——让他保持直立！——只有他的头和手臂伸出。呼吸困难可能要追溯到他十几岁时令人窒息的橱柜，或者可能是充血性心力衰竭。

The political battles were over, but not the ones related to health. He was never free from some physical plague or another. As an aging man he was so bothered with breathing problems that he had a special box built for himself to hold him upright when he was writing or speaking—or sleeping!—with only his head and arms protruding. The breathlessness might have dated back to the asphyxiating cupboard of his teenage years, or it might have been congestive heart failure.

傅立叶于 1830 年死于心脏病，仅比拿破仑晚了几年。他在巴黎著名的 Pére Lachaise 公墓的坟墓以波浪和埃及图案为特色（图 1.11）。他半身像的两侧是一朵百合花，茎很长，是波浪的轴心。每朵百合花的顶部都有一条戴着太阳圆盘的饲养眼镜蛇。如果埃及确实引起了傅立叶对热的痴迷，那么他的纪念碑的象征意义就是庆祝因果关系。

Fourier died of a heart attack in 1830, having outlasted Napoleon by only a few years. His grave in Paris’s famed Pére Lachaise Cemetery features the wave and Egyptian motifs (figure 1.11). To either side of his bust is a lily with a very long stem that is the axis of a wave. Each of the lilies is topped by a rearing cobra crowned with a solar disk. If Egypt did induce Fourier’s obsession with heat, then his monument’s symbology celebrates both the cause and effect.

两年后，年轻的格勒诺布洛瓦兹、埃及学的第一位教授让-弗朗索瓦·商博良跟随傅立叶来到拉雪兹神父。他的兄弟雅克-约瑟夫·商博良-菲雅克撰写了最早的傅立叶传记之一。傅立叶的挚友索菲·杰曼（Sophie Germain）——在谈到她时，他几乎不像是弗莱克斯——在他之后不久也被安葬在拉雪兹神父。

The younger Grenobloise, Jean-François Champollion, the first professor of Egyptology, followed Fourier to Pére Lachaise only two years later. His brother, Jacques-Joseph Champollion-Figeac, wrote one of the earliest biographies of Fourier. And Fourier’s dear friend Sophie Germain—he hardly seems frileux when it comes to her—was also buried in Pére Lachaise shortly after him.

日耳曼的最后一幕是为傅立叶的拉雪兹佩尔纪念碑捐款。泊松，他们俩的敌人，一直顽固不化，没有做出贡献。17

One of Germain’s final acts was contributing to the fund for Fourier’s Pére Lachaise monument. Poisson, enemy of them both and recalcitrant to the end, did not contribute.17

傅立叶的烦恼反映在他 1807 年的原创杰作的命运中，这本“回忆录”首先宣布了他的伟大成果。这份手稿消失了 160 多年，终于在国立桥梁和道路学院的图书馆中重新发现。这并不像最初看起来那么奇怪。这所学校是世界上最古老的土木工程学校，比傅立叶还要古老。18

Fourier’s troubles are reflected in the fate of his original 1807 masterpiece, the “memoir” that first announced his great result. The manuscript disappeared for over 160 years, finally rediscovered lying in the library of the École nationale des ponts et chaussées, the National School of Bridges and Roads. This is not as strange as it may first seem. The school is the oldest civil engineering school in the world, older than Fourier.18

天才的本质

The Nature of Genius

为什么傅立叶的好主意会遇到这样的阻力？正如他的批评者所看到的那样，问题在于：一个高度不规则的模式——比如一首任意的歌曲或绘画——怎么可能等同于高度规则的波的总和？

Why did Fourier’s great idea meet with such resistance? The problem, as his critics saw it, was this: How could a highly irregular pattern—like an arbitrary song or painting—be equivalent to a sum of highly regular waves?

在数学中，某些东西被证明或未被证明，或者它被证明是不可证明的。傅立叶本人的理论并没有达到这个真理的顶峰，尽管他的直觉非常合理。它落到了年轻的 Peter Gustav Lejeune Dirichlet 来填补剩余的数学空白。他于 1826 年来到巴黎，结识并钦佩傅立叶。在老人的指导下，狄利克雷坚定地确立了这一理论——十分严谨——并于 1829 年发表，也就是傅里叶生命的最后一年。19

In mathematics something is proved or it’s not, or it’s provably unprovable. Fourier himself didn’t attain this pinnacle of truth for his theory, although his intuition was remarkably sound. It fell to young Peter Gustav Lejeune Dirichlet to close the remaining mathematical gaps. He came to Paris in 1826 and met and admired Fourier. With the older man’s guidance, Dirichlet firmly established the theory—with full rigor—and published it in 1829, the last year of Fourier’s life.19

尽管如此，一些数学家仍然为傅里叶数学深奥的死水困扰。但工程师不是。在 1960 年代后期，我参加了斯坦福大学 Ron Bracewell 的一门非常有影响力的傅立叶技术基础课程。他特别努力强调傅里叶理论边缘的数学困难，并教导其适用性的严格限制。但他也强调了这些数学细节如何不适用于对现实世界现象的分析。或者更确切地说，他明确表示，现实世界属于狄利克雷等数学家所建立的界限。

Nevertheless, some mathematicians are still troubled by the esoteric backwaters of Fourier’s math. But engineers aren’t. In the late 1960s I took a remarkably influential foundation course on Fourier techniques from Ron Bracewell at Stanford. He took special pains to emphasize the mathematical difficulties at the edges of Fourier theory and to teach the strict limits of its applicability. But he also emphasized how those mathematical niceties don’t apply to analyses of real-world phenomena. Or rather he made clear that the real world falls within the limits established by mathematicians like Dirichlet.

数学家必须处理所有可能的模式，而不仅仅是我们在现实世界中实际发现的模式。数学家处理抽象，但工程师处理物理现实——热、光、声音、道路和桥梁以及图像。对于工程师来说，傅立叶的频率和幅度与他们描述的物理世界一样“物理”。如果大自然产生了一种模式，那么傅立叶的伟大想法几乎总是可以用来描述它。20

Mathematicians must deal with all possible patterns, not just the patterns we actually find in the real world. Mathematicians deal in abstracts, but engineers deal in physical realities—heat, light, sound, roads and bridges, and images. To engineers Fourier’s frequencies and amplitudes are as “physical” as the physical world they describe. If Mother Nature produces a pattern, then Fourier’s great idea almost always works to describe it.20

牛顿和爱因斯坦知道他们在谈论宇宙，他们的崇拜者也是如此。傅立叶没有或无法预见他伟大的音乐理念有多么普遍——当时其他人也没有。我们没有关于通过时间积累的天才或通过分支产生的天才的词。我们通常对天才的概念是终生的——无论是在成就方面还是在认可方面。

Newton and Einstein knew that they were addressing the universe, and so did their admirers. Fourier didn’t or couldn’t foresee how universal his great musical idea was—nor did others at the time. We don’t have a word for genius accumulated through time, or genius by ramification. Our usual notion of genius is local to a lifetime—both in accomplishment and in recognition.

然而，两个世纪以来，工程师们已经成功地大量使用了他的谐波理念，让我们感到舒适和享受。所有现代媒体都依赖于它。Digital Light 的像素和故事只是最新的例子之一。

Yet for two centuries engineers have made successful and massive use of his harmonic idea for our comfort and enjoyment. All of modern media depends on it. The pixel and the story of Digital Light is just one of the latest examples.

毫无疑问，傅立叶现在已经达到了他应得的位置，无论我们是否选择称其为天才。只剩下跨越两种文化的过道，让他在双方都知道和尊重。

Surely Fourier has now arrived at his deserved position, whether we choose to call it genius or not. It only remains to reach across the aisle separating the two cultures and make him known and respected on both sides.

撒布机

The Spreader

想法是这样的：数字可以忠实地代表模拟。离散的、分离的、尖的可以准确地表示光滑的、连接的和弯曲的。断续可以真实地代表未断续。如果现在这不让你感到惊讶，那么我希望很快就会引起你的惊讶，因为看起来我们可以丢弃大量的信息——实际上是无限的——而不会丢失任何东西。这是使数字光（以及数字声音）成为可能的关键思想。这是促成大数字融合以及现代世界的基本真理。

Here’s the idea: digital can faithfully represent analog. The discrete and separated and spiky can accurately represent the smooth and joined and curved. The broken continual can truthfully represent the unbroken continuous. If this doesn’t surprise you now then I hope to provoke your amazement soon, because it appears that we can throw away an astounding amount of information—an infinite amount, in fact—without losing anything. This is the key idea that makes Digital Light (as well as Digital Sound) possible. It’s the fundamental truth that enabled the Great Digital Convergence and hence the modern world.

正如波是代表傅立叶频率的形状一样，也有一个形状代表 Kotelnikov 的样本（见图 2.1）。我们很快就会发现它与像素的“形状”密切相关。数学家称其为sinc，而工程师则称其为重构滤波器。因为这两个名字都不直观，所以我把这个可爱的形状称为吊具。你很快就会明白为什么。

Just as the wave is the shape that represents Fourier’s frequencies, there’s a shape that signifies Kotelnikov’s samples (see figure 2.1). We’ll soon see that it’s intimately related to the “shape” of a pixel. Mathematicians call it a sinc, and engineers a reconstruction filter. Because both those names are nonintuitive, I call this lovely shape a spreader. You’ll see why shortly.

请注意，散布器类似于傅里叶波的波峰和波谷，它们逐渐被挤压，直到它们最终在任一方向上都消失。事实上，这正是它的本质。相关波在任何地方都有一个与中央驼峰一样高的振幅，其频率与吊具的上下摆动频率相同（图 2.2）。

Notice that a spreader resembles one of Fourier’s waves with its crests and troughs progressively squeezed until they’re ultimately reduced to nothing in either direction. In fact, that’s exactly what it is. The associated wave has an amplitude everywhere as high as the central hump, and a frequency the same as the spreader’s frequency of up and down wiggles (figure 2.2).

吊具来自数学，而不是现实世界，但它可能会让你想起一颗落在静水中的鹅卵石，涟漪向外辐射，它们的高度随着距离的增加而降低。像波浪一样，吊具永远在各个方向摆动。但是距中央驼峰一定距离的波峰是如此之低，以至于它们无关紧要。正如我们将看到的，这在现实世界中很重要。最早的图片我在正确的上下文中发现的吊具出现在 Vladimir Kotelnikov 1933 年的经典俄罗斯论文中（图 2.3）。2

The spreader comes from mathematics, not the real world, but it might remind you of a pebble dropped in still water, with ripples radiating outward, their height decreasing with increasing distance. Like a wave, a spreader continues to wiggle in each direction forever. But the crests at some distance from the central hump are so low that they don’t matter. That’s important in the real world, as we’ll see. The earliest picture of the spreader that I’ve found, in correct context, appears in the classic 1933 Russian paper by Vladimir Kotelnikov (figure 2.3).2

1960 年代初，我在电气工程系第一次了解到本章的好主意。我们听说它来自美国电气工程师的英雄 Harry Nyquist。他在 AT&T 传说中的贝尔实验室工作——我们都梦想有一天能成为那里。但是在 1960 年代后期，当计算机科学最终成为一门独立的学科时，就我而言，在斯坦福大学，这个伟大的想法变成了克劳德·香农的想法。当谈到所有数字化事物时，香农是我们的新英雄。他是第一个在印刷品中使用比特这个词的人。作为奈奎斯特的年轻同事，他也在贝尔实验室工作。3

I first learned this chapter’s great idea in the early 1960s in an electrical engineering department. We heard that it came from Harry Nyquist, an American hero to us electrical engineers. He worked in the fabled Bell Labs at AT&T—where we all dreamed to be someday. But in the late 1960s, when computer science finally emerged as a separate discipline, at Stanford in my case, the great idea became Claude Shannon’s. Shannon was our new hero when it came to all things digital. He was the guy who had first used the word bit in print. And he worked at Bell Labs too, as Nyquist’s younger colleague.3

但这只是美国版本，这就是斯蒂格勒的地名法则发挥作用的地方：“没有科学发现以其最初的发现者命名。” （顺便说一句，这条定律并不是由斯蒂格勒发现的。）所以在俄罗斯，毫不犹豫地，这个想法的全部功劳总是归科捷利尼科夫所有。在日本，功劳归功于染谷功。在英格兰，致埃德蒙·惠特克爵士。在德国，赫伯特·拉贝。想来想去，奈奎斯特出生在瑞典。只有香农是一个真正的蓝色，出生于密歇根州的美国人。香农标签只是粗暴的民族主义吗？所有这些人都击败了香农，除了染谷之外，他只遵循了几个月的想法。大概保留给第一个发现者的命名荣誉几乎是最后一个发现者——不管怎样，在美国。4

But that’s just the American version, and here’s where Stigler’s Law of Eponymy comes into play: “No scientific discovery is named after its original discoverer.” (This law, by the way, was not discovered by Stigler.) So in Russia, with no hesitation, full credit for the idea always goes to Kotelnikov. In Japan, credit goes to Isao Someya. In England, to Sir Edmund Whittaker. In Germany, Herbert Raabe. Come to think of it, Nyquist was born in Sweden. Only Shannon was a true-blue, Michigan-born American. Is the Shannon tag just crude nationalism? All these men beat Shannon to a version of the idea except for Someya, and he followed by only a few months. The naming honor presumably reserved for the first discoverer went instead to almost the last—in the United States anyway.4

尽管存在归属问题，但事实很清楚：这个伟大的想法——正如它在 Digital Light 中所使用的——首先由 Kotelnikov 在 1933 年清楚、清晰、完整地陈述和证明。西方人可能很难相信这样一个基本的想法是在斯大林的俄罗斯最糟糕的日子里发明的。在冷战期间，我们被教导说，俄罗斯科学即使不像李森科的生物学那样虚假，充其量只是一种衍生品或只是宣传。是时候热身一下 Kotelnikov 值得称赞的事实了。5

Despite the attribution tangle, the facts are clear: the great idea—as it’s used in Digital Light—was first clearly, cleanly, and completely stated and proved by Kotelnikov in 1933. Westerners may find it hard to believe that such a fundamental idea was invented during the worst days of Stalin’s Russia. We were taught during the Cold War that Russian science was, if not bogus like Lysenko’s biology, at best derivative or just propaganda. It’s time to warm up to the fact that Kotelnikov deserves the credit.5

谱系、论文和职位

Pedigree, Papers, and Positions

Vladimir Aleksandrovich Kotelnikov 于 1908 年出生在喀山，一座位于莫斯科以东约 500 英里的伏尔加河畔古城。很难想象还有一位科学家拥有如此完美的数学血统。他的曾曾曾曾祖父谢苗·科捷尔尼科夫是莱昂哈德·欧拉的学生，他是有史以来最伟大的数学家之一。傅里叶使用的一些数学可能来自欧拉。1757 年，谢苗成为圣彼得堡科学院的早期院士，该科学院由彼得大帝创立，现称为俄罗斯科学院。6

Vladimir Aleksandrovich Kotelnikov was born in 1908 in Kazan, an ancient city on the Volga about 500 miles straight east of Moscow. It’s difficult to think of another scientist with such an impeccable mathematical pedigree. His great-great-great-grandfather, Semyon Kotelnikov, was a student of Leonhard Euler, one of the greatest mathematicians who ever lived. Some of the mathematics that Fourier used might have come from Euler. In 1757 Semyon became an early academician of the St. Petersburg Academy of Sciences, founded by Peter the Great and now called the Russian Academy of Sciences.6

Vladimir 的祖父 Petr Kotelnikov 是喀山大学的数学家。列宁和托尔斯泰是那里最著名的两个学生，但列宁被开除，托尔斯泰也退学了。留下来的人中有著名的几何学家尼古拉·罗巴切夫斯基（Nikolai Lobachevsky）。他挑战了古希腊的欧几里得几何，提出它关于平行线的第五个假设不一定是正确的——当时这个惊人的想法最终在爱因斯坦的广义相对论中找到了归属。祖父彼得是罗巴切夫斯基的助手，然后是冠军。

Vladimir’s grandfather, Petr Kotelnikov, was a mathematician at the University of Kazan. Lenin and Tolstoy were two of the most famous students there, but Lenin was expelled, and Tolstoy quit. Among those who stayed the course was the famed geometer Nikolai Lobachevsky. He challenged the ancient Greek geometry of Euclid by suggesting that its fifth postulate about parallel lines was not necessarily true—a startling idea at the time that eventually found a home in Einstein’s general theory of relativity. Grandfather Petr was Lobachevsky’s assistant and then champion.

毫不奇怪，弗拉基米尔的父亲亚历山大·科捷尔尼科夫也是喀山大学的数学家。但当亚历山大决定离开喀山并举家搬到基辅担任新的教学职位时，弗拉基米尔的故事才真正开始。

Not surprisingly, Vladimir’s father Aleksandr Kotelnikov was also a mathematician at the University of Kazan. But when Aleksandr decided to leave Kazan and move his family to Kiev for a new teaching position, Vladimir’s story really began.

1914 年 8 月，当德国军队突破该市的防御工事时，Kotelnikovs 和 6 岁的弗拉基米尔抵达基辅。民众惊慌失措，从城里冲了出来，带着新来的人一起扫荡。他们以巨大的困难设法忍受了 850 英里的撤退到喀山。这个家庭已经卷入了第一次世界大战的第一次爆炸。这是决定弗拉基米尔生活和事业的许多战争中的第一场。7

The Kotelnikovs, with six-year-old Vladimir, arrived in Kiev on the very day in August 1914 when the German army broke through that city’s defenses. The population panicked and rushed from the city, sweeping the new arrivals with them. With enormous difficulty they managed to endure the 850-mile retreat to Kazan. The family had ridden right into the first blasts of World War I. It was the first of many wars that would define Vladimir’s life and career.7

他接下来的两场战争是 1917 年的十月革命和随后的红军和白人之间的内战。俄罗斯发生了翻天覆地的变化。年轻的弗拉基米尔也是——但改变他的并不是战争。在一片混乱中，他第一次听到了无线电广播。

His next two wars were the October Revolution of 1917 and the subsequent civil war between Reds and Whites. Russia was transformed. Young Vladimir was too—but it wasn’t the wars that changed him. In the midst of the chaos he heard, for the first time, a radio broadcast.

“它是如何工作的？” 他问父亲。

“How does it work?” he asked his father.

“这是你还无法理解的东西。”

“It’s something you can’t understand yet.”

这一挑战使他在 10 岁时专注于广播生活。在接下来的九个十年里，他大部分时间都在无线电和通信工程上度过，这一职业与苏联的兴衰相吻合。8

That challenge focused him, at age 10, on a life in radio. He would spend most of the next nine decades in radio and communications engineering, a career coincident with the rise, turmoil, and fall of the Soviet Union.8

“伟大的一年，可怕的一年，我们的主 1918 年，革命开始以来的第二年。” 米哈伊尔·布尔加科夫的《白卫队》就这样开始了，他对基辅可怕的、破坏性的、无政府主义的世界的描述。在第二次时机不佳的情况下，Kotelnikovs 恰好选择了这个不幸的时刻再次搬到基辅并亲身体验布尔加科夫的噩梦。教授煮肥皂，孩子们揭开窗帘缝线——任何可以用来驱走饥饿的东西。9

“Great was the year and terrible the Year of Our Lord 1918, the second since the Revolution had begun.” So begins Mikhail Bulgakov’s White Guard, his account of the terrible, destructive, anarchical world of Kiev. In their second case of bad timing, the Kotelnikovs picked exactly this unfortunate moment to move to Kiev again and experience Bulgakov’s nightmare firsthand. The professor boiled soap, and the children unraveled curtains for the thread—anything to sell to drive away the hunger.9

1924 年，亚历山大再次举家搬家，这次是搬到莫斯科。他在莫斯科高等技术学校获得了一个新职位。学校的一部分将很快变身为莫斯科动力工程学院（俄语为MEI），并成为世界领先的技术大学之一。把它想象成莫斯科理工学院。10

In 1924 Aleksandr moved his family yet again, this time to Moscow. He took a new position at the Moscow Higher Technical School. Part of the school would soon metamorphose into the Moscow Power Engineering Institute (MEI in Russian) and become one of the leading technical universities in the world. Think of it as the Moscow Institute of Technology.10

Vladimir，我们的 Kotelnikov，是 MEI 的第一批毕业生之一。他们最初不会接受他，因为他的知识渊源——他不是工人或农民的血统——但幸运的规则改变允许他进入，他从未离开过。他 1931 年的文凭是电气工程专业，专攻无线电。他将在该机构工作 75 年。

Vladimir, our Kotelnikov, was one of MEI’s first graduates. They wouldn’t accept him initially because of his intellectual pedigree—he wasn’t of worker or peasant stock—but a fortunate rule change allowed him to enter, and he never left. His 1931 diploma was in electrical engineering, with a specialty in radio. He would be with the institution for 75 years.

然后它发生了——他的奇迹年。1932 年，科捷尔尼科夫在没有明显监督的情况下独自创作了两篇论文，每篇论文都将让他永远载入工程史册。其中之一是“非线性滤波器理论”，我们不再关注。但另一个包含了他的好主意，即采样定理。他在 11 月提交了它，并于次年以没有希望的标题“论电子通信中的‘以太’和电线的传输能力”出版。

Then it happened—his annus mirabilis. The year 1932 saw Kotelnikov create two papers on his own, with no apparent supervision, each of which would have established him in the annals of engineering forever. One of them was “The Theory of Non-Linear Filters” and won’t further concern us. But the other contained his great idea, the Sampling Theorem. He submitted it in November, and it was published the next year with the unpromising title, “On the Transmission Capacity of ‘Ether’ and Wire in Electric Communications.”

当他把它提交给 MEI 的教员时，其中一位说：“看起来很正确，但听起来更像科幻小说。” 从无到有。尽管如此，他们还是批准了它，使他走上了通往 MEI 院长的学术轨道。11

When he submitted it to the MEI faculty, one of them said, “It appears correct, but sounds more like science fiction.” Something from nothing. Nevertheless, they approved it, launching him on an academic trajectory toward a deanship at MEI.11

1933 年，在 MEI 担任讲师后，科捷尔尼科夫还开始为人民通讯委员会（俄语为 NKS）工作。沟通是最重要的在战争中的重要性，因此布尔什维克在 1917 年 10 月红色夺取政权的那天创建了 NKS 也就不足为奇了。科捷利尼科夫在那里成为了一名通信工程师，并最终领导了自己的研究所。他总是一只脚在象牙塔里，另一只脚在现实世界的政治和战争中。12

In 1933, with a lectureship at MEI, Kotelnikov also began working for the People’s Commissariat for Communications (NKS in Russian). Communications are of utmost importance in war, so it’s no surprise that the Bolsheviks had created NKS on the very day they seized power in Red October 1917. Kotelnikov became a communications engineer there and eventually headed his own institute. He would always have one foot in the ivory tower and the other in the real world of politics and war.12

他正在路上，在两个著名的组织中拥有两篇重要的论文和职位。他准备在学术界和政府中迅速发展。在两条道路上，科捷尔尼科夫都必须像傅立叶那样与暴君共舞。

He was on his way, with two important papers and positions in two prestigious organizations. He was poised for rapid advancement in both academia and government. On both paths Kotelnikov would have to dance with tyrants, as Fourier had.

数字和模拟无穷大

Digital and Analog Infinities

让我们不要对无限这个词感到害羞。实际上有许多不同类型的无穷大，但我们这里需要的只有两种是数字类型和模拟类型。熟悉的时钟秒针波图（图 2.4）将有助于清楚区分。

Let’s not be shy about the word infinite. There are actually many different kinds of infinity, but the only two we need here are the digital kind and the analog kind. The familiar diagram (figure 2.4) of the clock’s second-hand wave will help make the difference clear.

您会记得，在秒针绕表盘转动的每一个周期中，钟面上的每分钟标记都会在波浪上出现一个圆形黑点。随着秒针在表盘周围移动，圆点永远向右展开。他们有多少人？好吧，你可以数一数——一、二、三，等等——但你必须永远数数。那就是数字无限。总是有另一个。出于显而易见的原因，数学家称其为可数无穷大。

You’ll recall that there’s one round black dot on the wave for every minute mark on the clock’s face during each and every cycle the second hand makes around the dial. As the second hand progresses around and around the dial, the dots unfurl to the right forever. How many of them are there? Well, you can count them—one, two, three, and so forth—but you’ll have to count forever. That’s digital infinity. There’s always another one. Mathematicians call it countable infinity, for the obvious reason.

第二种无限，模拟无限，并不那么容易。考虑波浪上的两个连续点。点之间的波上有多少点？答：有很多你都数不过来。模拟无穷远大于数字无穷——听起来很奇怪。数学家 Georg Cantor 证明了这是真的，这就是他的意思：

The second kind of infinity, analog infinity, isn’t so easy. Consider two successive dots on the wave. How many points are there on the wave between the dots? Answer: there are so many you can’t even count them. Analog infinity is larger than digital infinity—as strange as that sounds. The mathematician Georg Cantor proved it was true, and here’s what he meant:

在波浪的任意两点之间，总会有另一点。例如，有一点——在波浪上——在两点之间。现在想想那个中点和左边的两个原点。他们之间有道理吗？是的，总是——例如，它们之间的中间点。现在重复这个四分之一点和两个原始点的左边。以此类推，无穷无尽，正如表达式所说的那样。困难在于您永远无法将其划分得足够细以结束划分过程。换句话说，你永远无法到达一个甚至可以开始数数的地方。数学家喜欢称它为不可数无穷大，但我会坚持使用模拟无穷大。两者都有效：平滑的事物具有模拟或不可数的无限部分。离散事物具有数字或可数无穷大的部分。从深刻的意义上说，数字小于模拟——即使你使用很多点来表示平滑的东西。

Between any two points on the wave, there’s always another point on it. For example, there’s the point—on the wave—halfway between the two points. Now think of that midpoint and the left of the two original points. Is there a point between them? Yes, always—for example, the point halfway between them. Now repeat for this one-quarter point and the left of the two original points. And so on, ad infinitum, as the expression goes. The difficulty is that you can never divide finely enough to reach an end to the dividing process. In other words, you can never get to a place where you can even start counting. Mathematicians like to call it uncountable infinity, but I’ll stick with analog infinity. Both work: Smooth things have an analog or uncountable infinity of parts. Discrete things have a digital or countable infinity of parts. In a profound way, digital is lesser than analog—even if you use lots of dots to represent the smooth thing.

但 Kotelnikov 的好主意似乎是——令人惊讶的是——数字等同于模拟。数字化不会丢失任何东西。一个离散的数字事物可以忠实地代表一个平滑的模拟事物。图 2.5 显示了一段声音片段，或者说是沿着一条水平线穿过它的视觉场景的片段。Kotelnikov 的想法在这两种情况下都有效。底部的直线是零响度或零亮度级别——完全无声或完全黑暗。曲线是声音随着时间的推移而变化的响度，或者是当你沿着一条线向右移动时视觉场景的亮度变化。在任何一种情况下，大点都会突出显示片段中等距的点。我们将通过这个一维示例建立直觉，

But Kotelnikov’s great idea appears to be—and here’s the surprise—that digital is equivalent to analog. Nothing is lost by going digital. A discrete digital thing can faithfully represent a smooth analog thing. Figure 2.5 shows a snippet of sound, say, or of a visual scene along a horizontal line through it. Kotelnikov’s idea works in either case. The straight line along the bottom is the zero loudness or zero brightness level—completely silent or completely dark. The curve is the changing loudness of the sound as time passes, or the changing brightness of the visual scene as you move to the right along one line through it. In either case the big dots highlight equally spaced points along the snippet. We’ll build intuition with this one-dimensional example, then gradually extend to the two dimensions that a full visual scene actually requires—just as we did for Fourier waves in the first chapter.

图 2.6 是如果你忽略平滑曲线上的所有点，除了大点处的点。在这些点之间，我们现在只有一条直线，即零响度或零亮度级别。不难想象二维版本会是什么样子。想想水平和垂直间隔相等的不平整的钉床。它们的高度会根据作为视觉场景的相应光滑表面上的亮度而变化。钉床将是一个“表面”，除了钉子外，其他地方都为零。

Figure 2.6 is what you get if you omit all the points on the smooth curve except the ones at the big dots. Between these points all we have now is the straight line, the zero loudness or zero brightness level. It’s not hard to imagine what the two-dimensional version would look like. Think of an uneven bed of nails, equally spaced horizontally and vertically. Their heights would vary according to the brightnesses on a corresponding smooth surface that was a visual scene. And the bed of nails would be a “surface” that is zero everywhere except at the nails.

图 2.5 为模拟，图 2.6 为数字。后者中的尖峰很自然地称为模拟曲线的样本。在二维的钉床情况下，指甲是相应模拟表面的样本。Kotelnikov 的好主意说，我们不需要完全平滑的曲线来表示声音，也不需要完全平滑的表面来表示视觉场景。我们只需要样本。换句话说，第一张图中高亮点之间的点的模拟无穷大可以忽略不计！他似乎在说没有什么可以代表什么。怎么可能？答案当然隐藏在“似乎”中。

Figure 2.5 is analog, and figure 2.6 is digital. The spikes in the latter are called, naturally enough, samples of the analog curve. In the bed-of-nails case for two dimensions, the nails are the samples of the corresponding analog surface. Kotelnikov’s great idea says that we don’t need the full smooth curve to represent a sound or the full smooth surface to represent a visual scene. We need only the samples. In other words, the analog infinity of points between the highlighted points in the first figure can be ignored! He appears to say that nothing can represent something. How could that be? The answer lies hidden, of course, in the “appears to.”

您可能会想，如果您只是使用更多样本并将它们“足够接近”放置，它们将成为模拟声音曲线。许多人都有同样的直觉，即像素——无论它们是什么——间隔足够近将成为它们所代表的视觉场景。但这种直觉是错误的。你不能靠得足够近。你不能让数字无限达到模拟无限。你无法计算不可数的东西。然而，Kotelnikov 似乎说你可以。是什么赋予了？

You might imagine that if you simply used more samples and placed them “close enough” they would become the analog sound curve. It’s the same intuition many people have that pixels—whatever they are—spaced closely enough together would become the visual scene they represent. But this intuition is wrong. You can’t get close enough. You can’t make digital infinity ever reach analog infinity. You can’t count what’s uncountable. Yet Kotelnikov appears to say you can. What gives?

此外，他的想法是，在第二张图中，显示的点足够接近——通过采集更接近的样本，你不会获得任何优势，也不会获得额外的信息。你还困惑吗？我希望如此，因为我们正在触及问题的症结——以及它的优雅。

Furthermore, his idea says that in the second figure the dots shown are close enough—that you gain no advantage, no additional information—by taking more closely spaced samples. Are you puzzled yet? I hope so because we’re getting to the crux of the matter—and its elegance.

随着这些问题在空中颤抖，我们几乎已经准备好第一次通过 Kotelnikov 的好主意了。但首先让我们回顾一下傅立叶的想法，因为科捷尔尼科夫的想法依赖于它。傅立叶告诉我们，模拟声音或视野可以用波的总和来表示。图 2.7 显示了我们一直在使用的模拟片段的波总和中的一个波，为方便起见，在顶部重复（用点指定样本位置）。您可以看到片段中没有任何东西比这个波更快地上下摆动，所以这个波是频率最高的那个。片段的傅立叶和中的所有其他波都具有较低的频率，否则您会在片段中的某处看到更快的摆动。

With these questions quivering in the air, we’re almost ready for a first pass at Kotelnikov’s great idea. But first let’s revisit Fourier’s idea, since Kotelnikov’s depends on it. Fourier taught us that an analog sound or visual field can be represented by a sum of waves. Figure 2.7 shows one of the waves in the sum of waves for the analog snippet we’ve been using, repeated at the top (designating the sample locations with dots) for convenience. You can see that nothing in the snippet wiggles up and down any faster than this wave, so this wave is the one with the highest frequency. All the other waves in the Fourier sum for the snippet have lower frequencies, otherwise you would see a faster wiggle somewhere in the snippet.

这是 Kotelnikov 的好主意：如果您以最高傅立叶频率的两倍对平滑的东西进行采样，那么您总是可以仅使用样本准确地恢复平滑的东西。样本是离散的、不连贯的、彼此分离的——绝对不平滑。这是他想法的第一部分——伟大的采样定理——表示可以用数字脱节性代替模拟平滑度的部分。第二部分告诉我们如何从数字样本中实际恢复原始模拟。

Here’s Kotelnikov’s great idea: if you sample something smooth at twice its highest Fourier frequency, then you can always exactly recover the smooth something using only the samples. The samples are discrete, disjointed, separated from one another—definitely not smooth. This is the first part of his idea—the great Sampling Theorem—the part that says it’s possible to substitute digital disjointedness for analog smoothness. The second part tells us how to go about the actual recovery of the original analog from the digital samples.

科捷尔尼科夫站在巨人傅立叶的肩膀上。傅里叶频率捕捉模拟图像在视野中的变化速度。然后 Kotelnikov 的伟大想法告诉我们如何数字化表示傅立叶波。令人惊讶的是，对于变化最快的波的每个周期，只有两个样本就足够了。直观地说，它需要两个，因为一个样本代表该波的波峰部分，另一个样本代表波谷部分。

Kotelnikov stands on the shoulders of the giant Fourier. Fourier’s frequencies capture how fast an analog picture changes across a field of view. Then Kotelnikov’s great idea tells us how to digitally represent Fourier’s waves. Astonishingly, only two samples suffice for each cycle of the fastest changing wave. It takes two, intuitively, because one sample represents the cresting part of that wave and the other the troughing part.

像素

The Pixel

Kotelnikov 的视野样本在数字世界中有一个名字。我们称它们为像素。就在那里！这就是像素的定义。它与傅立叶和科捷尔尼科夫密切相关。Kotelnikov 的采样使 Digital Light 成为可能。

There’s a name in the digital world for Kotelnikov’s samples of a visual field. We call them pixels. There it is! That’s the definition of pixel. It’s intimately associated with both Fourier and Kotelnikov. Kotelnikov’s sampling is what makes Digital Light possible.

像素不是小方块！这可能会让您感到惊讶，因为它们经常以这种方式表示——事实上，许多人经常将像素与相邻的小方块等同起来——这可能是对新生数字时代最普遍的误解。像素化这个词甚至使这种误解制度化。

Pixels are not little squares! This may surprise you because very often they’re represented that way—so often, in fact, that many people equate pixels with little abutting squares of color—which is perhaps the most widespread misunderstanding of the nascent digital age. The word pixelation has even institutionalized the misconception.

事实上，像素没有形状。它们只是在规则网格上采集的样本——不均匀的指甲床。它们只存在于一点，因此它们没有范围、没有宽度、零维度。你看不到它们，它们也没有可见的颜色。它们只有一个代表灰色阴影的数字，或三个代表颜色的数字。正如我们将看到的那样，使用 Kotelnikov 的想法，从数字中恢复模拟，似乎给出了像素形状。

In fact, pixels have no shape. They’re just samples taken on a regular grid—the uneven bed of nails. They exist only at a point, so they have no extent, no width, zero dimensions. You can’t see them, and they have no visible color. They just have a number representing a shade of gray, or three numbers representing a color. It’s the recovery of analog from digital, using Kotelnikov’s idea, that appears to give a pixel shape, as we’ll see.

像素这个词本身必须为生存而战。最初，像素被称为许多其他东西——例如点、点阵列、光栅元素、图片点和图片元素。图片元素赢了，但随后发生了一场关于用速记表示该术语的斗争。多年来，IBM 和 AT&T 一直努力将其承包给pel。但是凭借他们的候选像素，1960 年代中期充满活力的年轻图像处理社区战胜了大公司的努力。事实上，在像我这样在动荡的 1960 年代成长起来的计算机图形学家中，​​禁止 Big Blue 和 Ma Bell 的pel是一种反主流文化的荣誉。理查德·里昂（Richard Lyon）仔细研究了变体词的使用方式和时间，在 1965 年加州理工学院喷气推进实验室的弗雷德·比林斯利（Fred Billingsley）的文件中发现了像素的最早出现，特别是在其图像处理实验室（见图 2.8，并注意KC 表示每秒千次循环）。根据里昂的说法， pel最早的公开使用来自 1967 年麻省理工学院教授威廉·施赖伯 (William Schreiber) 的一篇论文。

The word pixel itself had to fight for existence. Pixels were called many other things at first—spots, point arrays, raster elements, picture points, and picture elements, for example. Picture elements won, but then came a battle over representing the term in shorthand. For many years IBM and AT&T made an effort to contract it to pel. But with their candidate pixel, the vibrant young image-processing community of the mid-1960s triumphed over the efforts of the giant corporations. In fact, it was a point of counterculture honor among computer graphicists like me, who came of age in the tumultuous 1960s, to disallow Big Blue’s and Ma Bell’s pel. Richard Lyon, in a careful history of how and when the variant words were used, found the earliest appearance of pixel in a 1965 document by Fred Billingsley of Caltech’s Jet Propulsion Lab, specifically in its Image Processing Lab (see figure 2.8, and note that KC means thousand cycles per second). The earliest public use of pel, according to Lyon, came from a 1967 paper by the MIT professor William Schreiber.

许多专利在 1970 年代发布，包含像素或pel，每年像素专利的数量大大超过 pel 专利的数量。不出所料，这个时代的大多数pel专利都分配给了 IBM 或 AT&T 的贝尔实验室。如果 Nyquist 或 Shannon 为像素使用了一个词——他们没有——那肯定是pel，Ma Bell 的词。13

Many patents were issued in the 1970s containing pixel or pel, with the number of pixel patents greatly exceeding the number of pel patents each year. Most of the pel patents of this era were unsurprisingly assigned to IBM or AT&T’s Bell Labs. If Nyquist or Shannon had used a word for the pixel—they didn’t—it would surely have been pel, Ma Bell’s word.13

像素直接调用采样定理；采样和像素在出生时就被加入了。奇怪的是，声音样本没有像像素这样的词，尽管 Kotelnikov 的采样同样的技巧使数字声音成为可能。不幸的是，采样在音频中具有不同的含义。例如，在嘻哈音乐中，这意味着借用他人的整段音乐，价值几秒钟，然后将它们放在一起，或者将它们混合在一起。在这本书中，一个独特的词非常方便，所以我使用soxel（声音元素的缩写）作为声音样本的名称。

The pixel directly invokes the Sampling Theorem; sampling and pixels were joined at birth. Oddly, there’s no word like pixel for a sound sample, despite the fact that the same trick, Kotelnikov’s sampling, is what makes Digital Sound possible. Unfortunately, sampling has a different meaning in audio. In hip-hop, for example, it means to borrow whole chunks of other people’s music, several seconds worth, and put them together, or mash them up. A distinct word is quite handy in this book, so I use soxel—a contraction of sonic element—as the name of the sound sample.

名称游戏

Name Games

一个更严重的命名问题出现了，因为在今天的美国，Kotelnikov 的采样定理几乎总是归因于香农。如果你忽略世界其他地方，很容易看出原因。克劳德·香农在美国是一个响亮的名字。在他 1948 年的论文“通信的数学理论”中，他陈述了采样定理。在他 1949 年极具影响力和经典的论文“存在噪声中的通信”中，他陈述并证明了采样定理，其形式现在通篇使用数字世界，尤其是数字光。随着他在美国赢得许多最高奖项——例如，国家科学奖章和广大电气和电子工程师协会 (IEEE) 的荣誉奖章，他的地位不断提高。他是第一位获得 IEEE 信息理论学会同名克劳德 E. 香农奖的人，因为他创立的领域。在国际上，他获得了第一个京都奖，相当于数学科学的诺贝尔奖。14

A more serious naming issue arises because in the United States today, Kotelnikov’s Sampling Theorem is nearly always attributed to Shannon. If you ignore the rest of the world, it’s easy to see why. Claude Shannon is a towering name in America. In his 1948 paper, “A Mathematical Theory of Communication,” he stated the Sampling Theorem. In his massively influential and classic 1949 paper, “Communication in the Presence of Noise,” he stated and proved the Sampling Theorem in the form now used throughout the digital world, particularly in Digital Light. His stature grew as he won many top awards in the United States—for example, the National Medal of Science and the Medal of Honor of the vast Institute of Electrical and Electronic Engineers (IEEE). He was the first recipient of the eponymous Claude E. Shannon Award of the IEEE Information Theory Society, for the field he established. Internationally, he won the first Kyoto Prize, an equivalent of the Nobel for mathematical sciences.14

克劳德·埃尔伍德·香农于 1916 年 4 月 30 日出生于密歇根州的佩托斯基。他在附近的盖洛德长大，据说他在他家和朋友家之间的铁丝网围栏上制作了一条粗糙的电报线。他喜欢杂耍、密码和国际象棋。他是一个顽皮的天才。他骑着独轮车穿过麻省理工学院的大厅，在那里他获得了博士学位，后来又穿过了贝尔实验室的大厅，在那里他发明了信息论。他在一个盒子里制造了一台机器，它只做一件事：当你拨动侧面的开关时，盖子打开，伸出一只手臂，然后拨动开关。

Claude Elwood Shannon was born on April 30, 1916, in Petoskey, Michigan. He grew up in nearby Gaylord where he reputedly crafted a crude telegraph line from the barbed-wire fences between his house and a friend’s. He loved juggling, secret codes, and chess. He was a playful genius. He rode unicycles through the halls at Massachusetts Institute of Technology, where he got his PhD, and later through the halls at Bell Labs, where he invented information theory. He built a machine in a box that did one thing: when you toggled a switch on its side, the lid opened, an arm reached out, and toggled off the switch.

香农是密码学大师，他在 1945 年写了一篇名为“密码学数学理论”的机密论文。在第二次世界大战期间，他分析了用于富兰克林·D·罗斯福和温斯顿·丘吉尔之间安全通信的 X 系统。香农能够在数学上证明其加密方案是牢不可破的。香农也是存在噪音的通信领域的重量级人物，这实际上是他 1949 年论文的标题。他展示了如何在整个太阳系中发送数字信息，从而消除在途中因宇宙噪声而发生的损坏，从而揭示原始信息。香农的想法使得从好奇号一路可靠地发送火星电影成为可能流动站到你的笔记本电脑上——所以很自然地，香农的名字被附在了采样定理上。但这不是香农的主意。他自己也是这么说的。15

Shannon was a master of cryptography and wrote a classified paper in 1945 called “A Mathematical Theory of Cryptography.” During World War II he analyzed the X System that was used for secure communications between Franklin D. Roosevelt and Winston Churchill. Shannon was able to prove mathematically that its encryption scheme was unbreakable. Shannon was also a heavyweight in the field of communication in the presence of noise, the title of his 1949 paper in fact. He showed how to send digital messages across the solar system in such a way that corruption occurring en route from cosmic noise could be removed to reveal the original message. Shannon’s ideas made it possible to reliably send movies of Mars all the way from the Curiosity rover to your laptop—so it was natural that Shannon’s name was attached to the Sampling Theorem. But it wasn’t Shannon’s idea. He said so himself.15

“这是传播艺术中的常识，”他在 1949 年的论文中写道。“它以前曾由数学家以其他形式给出，但尽管它具有明显的重要性，但似乎并没有明确地出现在传播理论的文献中。” 16

“This is a fact which is common knowledge in the communication art,” he wrote in the 1949 paper. “It has been given previously in other forms by mathematicians but in spite of its evident importance seems not to have appeared explicitly in the literature of communication theory.”16

但是通信界的某个人早在香农之前就已经以正确的形式和完整的证明陈述了采样定理。Kotelnikov 早在 1933 年就这样做了，比这早了十多年。为什么香农不提他？也许是因为早期的论文是作为晦涩的俄罗斯会议论文集的一部分发表的，而香农根本不知道。但正如我们将看到的，他保持沉默的原因很难确定。他本可以在二战期间俄罗斯和美国共享的非常秘密的科学背景下了解科捷尔尼科夫的采样定理。但两国之间的冷战分裂会在那些年里，香农要么无法公开他对定理的了解，要么无法了解它。

But somebody from the communications world had stated the Sampling Theorem—in the right form and with the full proof—long before Shannon. Kotelnikov had done so in 1933, well over a decade earlier. Why didn’t Shannon mention him? Perhaps because the earlier paper had been published as part of obscure Russian conference proceedings, and Shannon simply didn’t know about it. But as we’ll see, the reason for his silence is hard to determine. He could have learned about Kotelnikov’s Sampling Theorem in the context of the very secret science Russia and the United States shared during World War II. But the bitter Cold War division between the countries would have prevented Shannon either from making his knowledge of the theorem public or from learning about it during those years.

这是两个并排的巨人（图 2.9）。1932 年，24 岁的科捷尔尼科夫在左边——留着更狂野的发型——比他证明采样定理早一年。右边是身材苗条的香农，年龄未定，但可能是 32 岁左右，接近他在 1949 年初证明的时间。两人都是各自国家数字通信的领导者——尤其是在存在噪音或加密的情况下的通信。两人都获得了最高奖项。两者都陈述并证明了今天在数字光中使用的采样定理。

Here are the two giants, side by side (figure 2.9). Kotelnikov is on the left—with the wilder hairdo—in 1932 at age 24, a year before he proved the Sampling Theorem. On the right is the slender Shannon at an undetermined age, but perhaps about 32, near the time of his proof in early 1949. Both men were leaders of digital communications in their respective countries—especially communications in the presence of noise or encryption. Both received the highest awards. And both stated and proved the Sampling Theorem as it’s used today in Digital Light.

我们不禁怀疑是否存在跨文化泄漏。奇怪的是，几年后，年轻的香农经常与科捷尔尼科夫的智力成就相提并论——但我没有发现任何证据表明香农知道科捷尔尼科夫的工作。无论如何，俄罗斯人将这个伟大的想法称为科捷尔尼科夫采样定理也就不足为奇了。我们不应该吗？17

We can’t help but wonder if there was cross-cultural leakage. Curiously, the younger Shannon often paralleled Kotelnikov’s intellectual achievements a few years later—but I’ve found no evidence that Shannon knew Kotelnikov’s work. In any case, it’s no surprise that Russians call the great idea Kotelnikov’s Sampling Theorem. Shouldn’t we?17

传播和添加

Spread and Add

Kotelnikov 想法的后半部分——伟大的采样定理——告诉我们如何从非平滑像素重建平滑图像，并准确地做到这一点。这令人惊讶的是，数字图像中似乎几乎不存在任何信息——在每对像素之间简单地省略了模拟的无穷多点。数字音频和每对索素也是如此。采样定理的后半部分告诉我们在哪里可以找到那些缺失的无穷大。

The second half of Kotelnikov’s idea—the great Sampling Theorem—tells us how to reconstruct a smooth picture from non-smooth pixels, and to do it accurately. The surprise about digital is that almost no information seems to exist in a digital image—an analog infinity of points has simply been omitted between each pair of pixels. The same goes for digital audio and each pair of soxels. The second half of the Sampling Theorem tells us where those missing infinities can be found.

以下是如何从数字中恢复模拟。用传播器传播每个像素，这是本章的基础形状。将结果相加。而已。采样定理告诉我们，这种扩展和相加过程准确地再现了像素之间缺失的无穷大！与所有伟大的定理一样，这样的推论一点也不明显。我们必须相信数学。

Here’s how to recover the analog from digital. Spread each pixel with the spreader, the shape fundamental to this chapter. Add up the results. That’s it. The Sampling Theorem tells us that this spreading and adding process accurately reproduces the missing infinities between the pixels! As with all great theorems, such deductions aren’t at all obvious. We have to trust the math.

运行示例将再次提供帮助，但为简单起见仅显示了两个 soxel（图 2.10），即两个中心点。（我们将很快将其扩展到像素。）回想一下，soxel 是模拟声音曲线的样本，其中曲线在零线上方的高度代表它的响度。因此，索素的高度代表了索素采样点处曲线的响度。右侧的索克索比左侧的索克索声音更小——更安静。

The running example will help again, but with only two soxels shown for simplicity (figure 2.10), the two central ones. (We’ll extend it to pixels shortly.) Recall that a soxel is a sample of an analog sound curve, where the height of the curve above the zero line represents its loudness. So the height of a soxel represents the loudness of the curve at just that point sampled by the soxel. The soxel at the right is less loud—quieter—than the one on the left.

首先，我们将传播我们在图 2.1 中看到了带有撒布器的左侧索素。回想一下，它以与原始片段的最高频率的傅里叶波相同的频率摆动。它在中央驼峰处的最大振幅（响度）代表音量一直向上。要进行撒播，请将左侧索素替换为撒播器的副本（图 2.11）。我想说的是，这会将 soxel 从无形状（无）传播到显示的形状（有）。它的最高点——中央驼峰的顶峰——与它所取代的索克具有相同的响度。两个索素显示为虚线。特别是，这清楚地表明了扩展器的高度（其最大响度）与左侧索克的高度相匹配。在本例中，响度为全音量的 80%。想象一下，您有一个用于调节响度的旋钮。

First we’ll spread the left soxel with the spreader, which we saw in figure 2.1. Recall that it wiggles at the same frequency as the Fourier wave with the highest frequency of the original snippet. Its maximum amplitude (loudness), at the central hump, represents the volume turned all the way up. To do the spreading replace the left soxel with a copy of the spreader (figure 2.11). I like to say that this spreads the soxel from no shape (nothing) to the shape shown (something). Its highest point—the peak of its central hump—has the same loudness as the soxel it replaces. The two soxels are shown dashed. In particular, this makes clear that the height of the spreader—its maximum loudness—matches that of the left soxel. In this example, that loudness is 80 percent of full volume. Imagine that you have a knob for that loudness adjustment.

现在将扩展器的另一个副本移动到右侧索素上方的位置（图 2.12）并旋转旋钮，直到其最大响度与该索素的响度相匹配，在本例中为全音量的 50%——因此散布第二个索素。

Now move another copy of the spreader into place over the right soxel (figure 2.12) and twiddle the knob until its maximum loudness matches that soxel’s loudness, in this example 50 percent of full volume—hence spreading the second soxel.

这是将这两个“spread soxels”相加的结果（图 2.13）。在每个水平位置，取那里散布的 soxels 的两个高度（浅灰色），从零响度线测量，并将它们加在一起以获得粗曲线上的一个点。

And here’s the result (figure 2.13) of adding these two “spread soxels” together. At each horizontal position take the two heights of the spread soxels there (light gray), measured from the zero-loudness line, and add them together to get a point on the bold curve.

到目前为止，我忽略了现实。吊具——本章的特色形状——在现实世界中并不存在。它无限广阔。它继续永远地左右波动。显然，现实世界的吊具不可能如此广泛，因此现实世界的吊具必须是理想吊具的近似值。

I’ve ignored reality so far. The spreader—this chapter’s featured shape—doesn’t exist in the real world. It’s infinitely broad. It continues to ripple left and right forever. Obviously, no real-world spreader can be that broad, so real-world spreaders by necessity have to be approximations of the ideal spreader.

一种流行、实用且非常精确的吊具称为立方吊具（图 2.14）。请注意它与理想吊具的中间部分有多相似，包括存在两个负向波瓣（低于零响度线）。在中心样本的左侧和右侧两个样本（即由撒布器传播的那个）之外，立方撒布器在任何地方都为零。换句话说，它具有有限的宽度，因此它可以存在于现实世界中。

A popular, practical, and remarkably accurate spreader is called a cubic spreader (figure 2.14). Note how closely it resembles the middle part of the ideal spreader, including the presence of two lobes going negative (below the zero-loudness line). The cubic spreader is zero everywhere beyond two samples left and two right of the central sample—the one spread by the spreader. In other words, it has a finite width, so it can exist in the real world.

到目前为止，我所描述的是一维传播。声音仅在一维（时间）上变化，因此它是索素的准确图片，但不是像素的准确图片。像素扩展器必须在二维中扩展，因为图像在二维（空间）中扩展，通常称为水平和垂直。像素散布器必须将每个钉子（像素）散布在钉床中，以便每个“散布像素”都有助于我们可以看到的二维表面。您可以认为前面的图准确地显示了像素散布器在水平维度上的横截面，请记住，垂直维度上的横截面是完全相同的。但我们可以做得更好。

What I’ve described so far is one-dimensional spreading. Sound varies in one dimension only (of time) so it’s an accurate picture for soxels, but not for pixels. A pixel spreader must spread in two dimensions since images extend in two dimensions (of space), often called horizontal and vertical. A pixel spreader must spread each nail (pixel) in a bed of nails so that each “spread pixel” contributes to a two-dimensional surface that we can see. You can think of the preceding figures as accurately showing the cross section of a pixel spreader in the horizontal dimension, keeping in mind that the cross section in the vertical dimension is exactly the same. But we can do better.

图 2.15 显示了一个完整的二维像素扩展器。将断头台从其最亮的地方扔下——这样刀片就可以将这座小山切成两半，从山顶到底部，如图所示。然后显示的出血边缘正是上面的立方分布器。如果你把小山切成两半，你会看到同样的边缘。由于这个像素散布器在每个维度上都是一个立方散布器，所以它被称为双三次散布器。它是Adobe Photoshop（最古老和最流行的像素应用程序之一）中内置的那种，用于更改图片的大小。

Figure 2.15 shows a full two-dimensional pixel spreader. Drop a guillotine through its brightest point—so that the blade slices the little mountain in half, from its peak to its base as shown. Then the bleeding edge revealed is exactly the cubic spreader above. You’d see the same edge if you cut the little mountain in half the other way. Since this pixel spreader is a cubic spreader in each dimension, it’s called a bicubic spreader. It’s the kind built into Adobe’s Photoshop (one of the oldest and most popular pixel apps) for changing the size of a picture.

所以现在让我们展开并添加一行像素，但让我们使用这个双三次展开器而不是理想的展开器来实现。为了简单和一致，我们将在横截面中执行此操作。但是想象一下，每个吊具实际上是一座小山，就像我们刚刚看到的那样——不平坦的钉床在另一个维度上延伸，每个钉子上都有一座小山。首先，我们散布像素（图 2.16）。

So now let’s spread and add a row of pixels, but let’s do it by using this bicubic spreader instead of the ideal one. For simplicity and consistency, we’ll do this in cross section. But imagine that each spreader is actually a little mountain like the one we’ve just seen—and that the uneven bed of nails extends in the other dimension with a little mountain at each of its nails. First, we spread the pixels (figure 2.16).

然后我们将它们全部加起来，以获得无处不在的亮度。图 2.17 显示了沿着通过视觉场景的一条水平线的结果。粗体亮度曲线重建了原始模拟片段，为了便于比较，我在其下方重复了该片段。它并不完美——只是一个近似值——因为我们使用了一个不太理想的吊具，但它很接近。与原始钥匙相比，它就像一把磨损或重复的钥匙。双方的牙齿都打开了锁。

Then we add them all up to get a resulting brightness everywhere. Figure 2.17 shows that result along one horizontal line through a visual scene. The bold brightness curve reconstructs the original analog snippet, which I repeat just below it for ease of comparison. It’s not perfect—just an approximation—since we’ve used a less than ideal spreader, but it’s close. It’s like a worn or duplicate key compared to the original. The teeth of either open the lock.

你可以想象，这些数字——对于十几个像素来说已经很复杂了——对于比如一百万个像素来说会变得非常复杂。重要的一点是每个步骤都很简单：将吊具移动到位，调整其亮度，然后将其添加到其他吊具上。我们有一种野兽，它可以一遍又一遍地重复简单的步骤，而不会感到困惑或无聊——当然是计算机。这就是为什么接下来的两章是关于计算机及其惊人的能力来放大（用大写字母A！）一个微不足道的人类步骤——比如说——将一个像素扩展一百万或十亿倍，而且做得非常快，更不用说将所有这些散布像素加起来。一个加法很简单——即使是人类也能做到——但它需要重复一百万或十亿次。这就是为什么我们需要计算机来真正实现数字光，尽管 Kotelnikov 和傅立叶的数学表明我们原则上可以做到。

You can imagine that these figures—already complex for just a dozen or so pixels—would become hopelessly complex for, say, a million pixels. The important point is that each step is simple: move a spreader into place, adjust its brightness, and add it to the others. We have a beast that can repeat an easy step over and over and not become confused or bored—the computer, of course. That’s why the next two chapters are about the computer and its marvelous ability to Amplify (with a capital A!) a puny human step—spreading one pixel, say—by a millionfold or billionfold, and to do it very fast, not to mention adding up all those spread pixels. One addition is simple—even a human can do that—but it needs to be repeated a million or a billion times. That’s why we need computers to actually realize Digital Light even though Kotelnikov’s and Fourier’s math shows that we can do it in principle.

让我们再次穿越魔法。有人在很远的地方——比如在北京——拍摄了一个事件的视频，也许是一场歌剧，或者是一个声音的录音，也许是一场音乐会，然后把大部分都扔掉了，只保留了一些样本。也许这些像素或索素流是从距离地球数千英里的通信卫星上反弹回来的，或者通过万维网运送给旧金山的你。在您访问它之前，该流可能已经存在于计算机文件或 DVD 中的某处多年。在穿越时空的所有旅程中，原始的模拟无限消失了。它所存在的只是你看不到或听不到的像素或索像素。即使您最终将它们下载到您的计算机或手机上，模拟无穷大仍然丢失，您仍然无法看到或听到它们。然后神奇的事情发生了：你决定看像素或听索素。在那一刻，我刚刚描述的传播和添加——重建——发生了，你会看到北京人在许多英里或几年前所做的完全模拟的无限信息。

Let’s step through the magic again. Someone took a video of an event, an opera perhaps, or an audio recording of a sound, maybe a concert, somewhere far away—in Beijing say—and threw most of it away, preserving only samples of it. Perhaps that stream of pixels or soxels was bounced off a communications satellite thousands of miles above the Earth or shipped across the World Wide Web to you in San Francisco. The stream may have lived in a computer file or on a DVD somewhere for years before you accessed it. During all that journey through space and time, the analog infinity of the original was missing. All that existed of it was pixels or soxels you couldn’t see or hear. Even when you finally downloaded them onto your computer or cellphone, the analog infinity was still missing, and you still couldn’t see or hear them. Then the magic happens: You decide to look at the pixels or listen to the soxels. There, at that moment, the spreading and adding I just described—the reconstruction—happens and you witness the full analog infinity of information that the person in Beijing did, many miles or years ago.

丢失的无穷大只有在屏幕或扬声器上显示时才会重新出现——而不是之前。我们认为我们正在携带图像或声音，但我们不是。我们正在运输一种高度压缩的形式。上个世纪的旧媒体——例如电影和视频——确实保留了从源头到通信通道到显示器的整个路径上的模拟无穷大。新媒体将其缩写为数字样本，并且仅在最后一刻重新发明原始模拟图像或声音。

The missing infinity only reappears when it’s displayed on the screen or the speaker—and not before. We think we’re carrying around the picture or the sound, but we aren’t. We’re transporting a highly compressed form of it. The old media of the last century—film and video, for example—did preserve the analog infinity all along the path from the source through the communication channel to the display. The new medium abbreviates it to digital samples and only reinvents the original analog picture or sound at the last possible moment.

所以这就是采样定理的秘密，以及为什么它不是真的一劳永逸。我们使用的任何吊具，无论它是否理想，都是模拟形状：它具有模拟无限点。通过在每个像素上放置一个扩展器（无穷大的小斑点）并将结果相加，我们有效地在任何地方重新引入了模拟无穷大。像素之间明显的空白被像素扩展器中携带的东西所覆盖。这是一个非常巧妙的技巧。采样定理重新包装了无穷大。数学的力量将我们从直觉的束缚中解放出来，这在科捷尔尼科夫的伟大想法中得到了充分发挥。

So here’s the secret of the Sampling Theorem, and why it’s not really something for nothing. Any spreader that we use, whether it’s the ideal one or not, is an analog shape: it has an analog infinity of points. By putting a spreader at each pixel—little blobs of infinity—and adding up the results, we effectively reintroduce analog infinity everywhere. The apparent nothing between pixels is covered by the something carried in the pixel spreader. It’s a very neat trick. The Sampling Theorem repackages infinity. The power of mathematics to free us from the bounds of our intuition is at full play in Kotelnikov’s great idea.

像素不是正方形

Pixels Are NOT Square

像素在被散布器散布之前是没有形状的。我经常将它们绘制为侧视图或正视图，以方便绘图，作为适当高度的尖峰。图 2.18（左）显示了运行示例的中心两个像素。

Pixels have no shape until they’re spread by a spreader. I’ve often drawn them for pictorial convenience as a side view, or in elevation, as spikes of the appropriate height. Figure 2.18 (left) shows the center two pixels of the running example.

表示像素的另一种方法是从上方查看它们，可以这么说，这样您就可以自然地“看到”它们。但是我们看不到像素。它是我们看到的散布像素。那么，从上方看，散布像素是什么样子的——也就是说，当我们实际查看像素时？图 2.18（中）是使用 Photoshop 的双三次散布器散布的两个像素的图片。这是两座小山，从它们的山峰上往下看。左侧较浅的散布像素几乎看不到，但它就在那里，与较暗的像素重叠。

Another way to represent pixels is by looking at them from above, so to speak, so that you can “see” them in a natural way. But we can’t see pixels. It’s spread pixels that we see. So what does a spread pixel look like from above—that is, as we actually view pixels? Figure 2.18 (middle) is a picture of the two pixels, as spread with Photoshop’s bicubic spreader. It’s two little mountains as seen from above their peaks, looking down. The lighter spread pixel on the left is barely visible, but it’s there, overlapped by the darker one.

两个展开的像素肯定不像图 2.18（右）中的邻接正方形。

The two spread pixels certainly don’t look like the abutting squares in figure 2.18 (right).

如果您必须为像素指定形状，则使用散布像素的斑点图像作为指导，并记住与相邻像素的重叠。但真的，真的，像素没有形状。这是一个直径为零的点。只有散布像素有形状。如果它是一个你可以看到的像素——如果它有一个可见的形状——那么它一定是一个扩展的像素。

If you must give a shape to a pixel, then use the blobby image of spread pixels as your guide and remember the overlap with neighboring pixels. But really, truly, a pixel has no shape. It’s a point with zero diameter. Only spread pixels have shape. If it’s a pixel you can see—if it has a visible shape—then it must be a spread pixel.

很容易理解小正方形概念的来源。常见的应用程序支持这种错觉。我在 Photoshop 中制作了这个小图 2.19（左，仔细看）。它是白色背景上的 14 个灰色像素簇，太小而无法轻易看到。当然，您实际上并没有在这里看到 14 像素。您会看到 14 个散布像素。

It’s easy to understand where the little square notion comes from. Common apps support the illusion. I made this—tiny—figure 2.19 (left, look carefully) in Photoshop. It’s a cluster of 14 gray pixels on a white background, and too small to see easily. Of course, you’re not actually seeing 14 pixels here. You’re seeing 14 spread pixels.

几乎所有像素应用程序都有一个称为缩放的功能，或者一些变体。缩放旨在向您展示图片的近距离外观，就好像它是通过放大镜观看的一样。图 2.19（中）显示了使用 Photoshop 的 Zoom In 将 14 个像素放大 16 倍的结果。

Nearly all pixel apps have a feature called Zoom, or some variant. Zoom purports to show you what a picture looks like up close, as if it were viewed through a magnifying glass. Figure 2.19 (middle) shows the result of using Photoshop’s Zoom In to make the 14 pixels 16 times larger.

那么像素肯定是小方块，对吧？一点也不。Zoom 使用了一种快速而肮脏且不准确的技巧，让您认为它在“放大”。缩放只是将每个像素水平重复 16 次，然后将每行 16 个相同的副本垂直重复 16 次。结果是每个原始像素都被替换为 16 x 16 的相同颜色像素的正方形阵列。所以每个数组看起来——真是令人惊讶——就像那个颜色的正方形。但每个正方形绝对不是一张原始像素特写的图片。这是一张200多张的照片每个原始像素的（扩展）像素，排列成一个正方形。Zoom 技巧在过去计算机速度慢的糟糕时代很有用，但在现代世界中，它只是具有误导性。是时候摒弃像素是小方块的概念了。从来都不是真的。

Surely then the pixels must be little squares, right? Not at all. Zoom uses a quick-and-dirty—and inaccurate—trick to make you think it’s “magnifying.” Zoom simply repeats each pixel 16 times horizontally, and then repeats each row of 16 identical copies 16 times vertically. The result is that each original pixel is replaced with a 16 by 16 square array of pixels of the same color. So each array looks—what a surprise—like a square of that color. But each square is definitely not a picture of one of the original pixels close up. It’s a picture of more than 200 (spread) pixels for each original pixel, arranged in a square. The Zoom trick was useful back in the bad old days of slow computers, but in the modern world it is simply misleading. It’s time to scotch the notion that pixels are little squares. It never was true.

图 2.19（右）向您展示了原始图片的真实样子，当 Photoshop 将 Photoshop 适当放大 16 倍时（具有称为图像大小的功能）。采样定理的后半部分解释了为什么你会看到斑点，每个斑点的中心都是原始灰色，但在各个方向都迅速软化为更白的灰色阴影，部分重叠相邻的斑点。

Figure 2.19 (right) shows you what the original picture looks like really, when Photoshop does a proper magnification by 16 (with a feature called Image Size). The second half of the Sampling Theorem explains why you see blobs, each with the original gray in the center but softening rapidly in every direction to whiter shades of gray, partially overlapping neighboring blobs.

大恐怖及其暴君

The Great Terror and Its Tyrants

斯大林的大清洗——大恐怖——从 1934 年持续到 1939 年。它是由克格勃的前身、臭名昭著的内务人民委员部（NKVD）实施的。许多人在臭名昭著的表演审判中被判处死刑，数十万甚至数百万俄罗斯人被送往古拉格群岛的战俘营——这些岛屿。是索尔仁尼琴向我们讲述了古拉格和正在发生的“在人们记忆中的历史上前所未有”的悲剧。18

Stalin’s Great Purge—the Great Terror—lasted from 1934 to 1939. It was implemented by the notorious NKVD (People’s Commissariat for Internal Affairs), predecessor of the KGB. Many were condemned to death in the infamous show trials, and hundreds of thousands, perhaps millions, of Russians were sent to the prison camps—the islands—of the Gulag Archipelago. It was Solzhenitsyn who taught us about the Gulag and the unfolding tragedy that “was unprecedented in remembered history.”18

Kotelnikov的两个暴君从这种恐怖中出现。大恐怖的权威历史学家罗伯特·康奎斯特（Robert Conquest）将领导清洗的四位强大的年轻斯大林主义者描述为“特别凶残”。其中有科捷利尼科夫的暴君格奥尔基·马林科夫和拉夫伦蒂·贝利亚。19

Two tyrants for Kotelnikov emerged from this horror. Robert Conquest, the definitive historian of the Great Terror, described four powerful young Stalinists who led the purge as “particularly murderous.” Among them were Kotelnikov’s tyrants Georgi Malenkov and Lavrenti Beria.19

马林科夫于 1941 年 2 月在莫斯科向第十八次党代会的报告拉开了又一次清洗的序幕，即斯大林的秘密清洗，仅次于大清洗。马林科夫透露，工业产能锐减，无能横行。因此，就在希特勒于 6 月入侵俄罗斯之前，数十个人民委员会（俄语中的 NK——我们可能称之为联邦局或国家部门）被摧毁，尤其是通讯人民委员会 (NKS)。整个 NKS 都解散了——除了 Kotelnikov 的实验室，因为它在战时的重要性而没有受到影响。20

Malenkov’s report to the Eighteenth Party Conference in Moscow in February 1941 kicked off yet another purge, Stalin’s Secret Purge, second only to the Great Purge. Malenkov revealed that industrial capacity was plummeting, and incompetence was rampant. So just before Hitler invaded Russia in June, dozens of the People’s Commissariats (NKs in Russian—what we might call Federal Bureaus or National Departments) were decimated, including in particular the People’s Commissariat for Communications (NKS). The entire NKS was disbanded—except for Kotelnikov’s lab, which was left untouched because of its wartime importance.20

二战期间，斯大林领导了一个管理国家的五人委员会：马林科夫和贝利亚都是成员。贝利亚还在副手维克托·阿巴库莫夫的协助下管理内务人民委员会，他是科捷利尼科夫的第三位暴君。当斯大林创建 SMERSH（是的，你们伊恩弗莱明的粉丝，这个反情报组织确实存在）时，他让阿巴库莫夫负责。

During World War II, Stalin headed a committee of five that ran the country: Malenkov and Beria were both members. Beria also ran the NKVD with the assistance of a deputy, Viktor Abakumov, a third Kotelnikov tyrant. When Stalin created SMERSH (yes, you Ian Fleming fans, this counterintelligence organization really existed), he put Abakumov in charge.

二战后马林科夫变得强大，起初仅次于斯大林。斯大林控制马林科夫和贝利亚的技巧是将阿巴库莫夫提升到更高的位置。因此，贝利亚和阿巴库莫夫掌管着控制古拉格奴隶劳改营的国家机器。他们在列宁格勒组织了大规模处决他们的对手，并将数千人送往集中营。1953年斯大林去世后，马林科夫成为苏联总理。

After World War II Malenkov became powerful, second only to Stalin at first. Stalin’s technique for keeping Malenkov and Beria under control was to raise Abakumov to a still higher position. So Beria and Abakumov ran the state apparatus that controlled the Gulag slave labor camps. They organized massive executions of their rivals in Leningrad and had thousands more sent to the camps. And when Stalin died in 1953, Malenkov became Premier of the Soviet Union.

但马林科夫对科捷尔尼科夫的重要性主要在于他的妻子瓦莱里娅·戈卢布佐娃。她将成为科捷尔尼科夫职业生涯中的决定性力量——他的保护者。21

But Malenkov’s importance for Kotelnikov lay principally in his wife, Valeriya Golubtsova. She would be the defining force in Kotelnikov’s career—his protectress.21

戈卢布佐娃拥有一流的革命血统。她的母亲奥尔加是涅夫佐罗夫姐妹中的一员——齐奈达、索菲亚、阿夫古斯塔和奥尔加——早在十月革命之前，她们年轻时都是列宁核心圈子的一部分。其中两个姐妹嫁给了有关系的男人：Zinaida 和她的丈夫 Gleb Krzhizhanovsky 在求爱和早婚期间与列宁和他的妻子一起流亡西伯利亚。格列布是列宁的朋友，也是革命运动中最年长的人物之一，自 1893 年以来一直是党员。索菲亚的丈夫谢尔盖·谢斯特宁 (Sergei Shesternin) 是列宁信任的拥有绝密布尔什维克财务的人。22

Golubtsova had a first-rate revolutionary pedigree. Her mother, Olga, was one of the Nevzorov sisters—Zinaida, Sophia, Avgusta, and Olga—who were part of Lenin’s inner circle when they were all young, long before the October Revolution. Two of the sisters married men with connections: Zinaida and her husband Gleb Krzhizhanovsky had spent exile in Siberia with Lenin and his wife during their courtships and early marriage years. Gleb was Lenin’s friend and one of the oldest figures in the revolutionary movement, a Party member since 1893. Sophia’s husband, Sergei Shesternin, was the man Lenin trusted with top-secret Bolshevik finances.22

Golubtsova 非常出色——她自己是莫斯科动力工程学院 (MEI) 的一名学生，当时女性工程师几乎闻所未闻。她也很强大，很难，而且很酷。作为一个坚定的组织者，她成为了 MEI 的长期主管，并将其变成了一个学术强国。当她为她的机构找到一位知识明星——比如科捷尔尼科夫——时，她可以而且会确保他留在那里，而不是在古拉格。如果她自己不完全是一个暴君，她非常有效地利用她的暴君身份得到了她想要的东西。23

Golubtsova was brilliant—a student at Moscow Power Engineering Institute (MEI) herself when women engineers were almost unheard of anywhere. She was also powerful, difficult, and cool. A determined organizer, she became MEI’s longtime director and turned it into an academic powerhouse. When she spotted an intellectual star for her institution—such as Kotelnikov—she could and would make sure he stayed there and not in the Gulag. If not exactly a tyrant herself, she used her tyrant-in-law status very effectively to get what she wanted.23

数码灯的工作原理

How Digital Light Works

Kotelnikov 制定了采样理论，并设计了将像素散布到图片中的理想方式。但他当时不知道这些想法将如何实际用于实践。一方面，现代数字世界有一个巨大的实际假设：实际设备，而不仅仅是理论波，将在另一端进行必要的传播和添加，以恢复您眼睛所需的模拟光。

Kotelnikov worked out the theory of sampling, and he designed the ideal way to spread a pixel into a picture. But he had no idea at the time of the way those ideas would actually come to be used in practice. For one thing, there’s a huge practical assumption in the modern digital world: an actual device, and not just a theoretical wave, will be at the other end to do the spreading and adding necessary to restore the analog light your eye requires.

Digital Light 之所以有效，是因为它假设丢失的信息会神奇地出现——以适当的像素散布器的形式——在过程中的某个地方，在显示的那一刻。数字声音的工作方式相同。它假设稍后某处会有一个设备将 soxels 传播到您耳朵所需的原始模拟声音中。声音的显示是一个扬声器系统和为其供电的放大器。实际上，显示是从样本中重建模拟原件（大脑需要的形式）的行为。

Digital Light works because it assumes that the missing information will magically appear—in the form of an appropriate pixel spreader—somewhere in the process, at the moment of display. Digital Sound works the same way. It assumes there’ll be a device somewhere later that will spread the soxels into the original analog sound that your ear requires. The display of sound is a speaker system and the amplifier that feeds it. Display, in fact, is the act of reconstructing the analog original—the form your brain needs—from samples.

显示是现代世界的一个基本假设。伟大的数字融合取决于这样一个事实，即相同的像素——图像的相同抽象表示——可以通过无数种技术以无数种方式显示。

Display is a fundamental assumption of the modern world. The Great Digital Convergence depends on the fact that the same pixels—the same abstract representation of an image—can be displayed in myriad ways by myriad technologies.

如果您用手机拍照，然后在电脑屏幕上查看，您可能会看到两种不同的像素分布。如果您在纸上查看它，毫无疑问您正在目睹不同的传播。唯一不变的是显示器后面的实际像素，由显示器重建的不可见像素，一个带有数字的点 - 如果它是彩色的，则为三个数字。Digital Light 的一项不为人知的成就是将不同显示技术的变幻莫测——毫无疑问还会有更多——减少到称为显示器的薄硬件层及其称为显示驱动程序的软件。每个制造商都有自己的显示器和驱动程序。除了像素本身，我们用户不必担心任何事情。我们的显示器进行传播。

If you take a picture with your cellphone and then look at it on your computer screen, chances are that you’re seeing two different pixel spreads. If you view it on paper, there’s no question that you’re witnessing a different spread. The only thing that’s constant is the actual pixel behind the display, the invisible one that’s reconstructed by the display, a point with a number attached—or three numbers if it’s colored. One of the unsung accomplishments of Digital Light is the reduction of the vagaries of different display technologies—and many more to come, no doubt—to a thin layer of hardware called the display and its software called a display driver. Each manufacturer makes its own displays and drivers. We users never have to worry about anything but the pixels themselves. Our displays do the spreading. This is the Great Digital Convergence at work.

这是一个分裂的融合。它在像素的创建和显示之间形成了一个楔子。在创意空间中，像素是简单、纯粹和通用的。在 Display Space 中，橡胶与道路相遇。现实世界技术的所有肮脏问题都与传播创意空间提供的像素的问题有关。在这里，数千家制造商的工程专业知识专注于在实际物理材料（等离子、液晶、荧光粉、墨水）的现实限制范围内对理想吊具进行高质量近似。我们用户需要担心的只是创意空间的柏拉图像素。这就是为什么我敦促传播可以感知的像素不应与不能感知的实际像素相混淆。让我们不要将稀薄的展示空间与广阔的创意空间混为一谈。

It’s a convergence that divides. It drives a wedge between the creation and display of pixels. In Creative Space, pixels are simple, pure, and universal. In Display Space, the rubber meets the road. All the nastiness of actual real-world technology comes to bear on the problem of spreading the pixels that are supplied from Creative Space. Here’s where the engineering expertise of thousands of manufacturers is focused on high-quality approximations of the ideal spreader within the real-world limitations of actual physical stuff—plasmas, liquid crystals, phosphors, inks. All we users need to worry about are the platonic pixels of Creative Space. That’s why I’ve urged that the spread pixel, which can be perceived, should not be confused with the actual pixel, which cannot. Let’s not conflate thin Display Space with vast Creative Space.

显示器制造商通常将显示器中的小发光元素称为“像素”。但这会将扩展像素与像素混淆。这些应该称为显示元素，而不是像素。这些显示元件之一的辉光——发射的照明，在中心高，从中心逐渐减小到零——是特定显示设备的散光器。一个像素进入显示元素，一个扩展像素出来，这就是你实际看到的。一般来说，发光形状是制造商独有的，甚至在同一家公司的产品之间也会有所不同。

Often display manufacturers call the little glowing elements in their displays “pixels.” But this confuses the spread pixel with the pixel. These should be called display elements, not pixels. The glow—the emitted illumination, high at the center, diminishing to zero away from center—of one of these display elements is the spreader for a particular display device. A pixel goes into the display element, a spread pixel comes out, and that’s what you actually see. Generally, the glowing shape is unique to the manufacturer and can even vary from product to product at the same company.

所以，这就是 Digital Light 的工作原理：它从现实世界中提取像素，甚至从火星或金星——或者从我们将看到的计算机内部的虚幻世界——中提取像素并将它们发送到各处。这些像素只携带离散的位置和强度（或颜色）信息。似乎来自像素的实际光仅在您用眼睛看到它之前的最后一刻由显示设备产生 - 而不是在世界各地或我们姐妹行星之间的任何中间位置。原始画面的流畅度只会重新出现在显示屏中。如果像素的原始创建做得很好，根据采样定理，并且如果将这些像素重建为模拟显示器也做得很好，也根据采样定理，那么您所看到的就是原始的准确表示。我们做从无到有。像素是虚无——它没有维度——而显示器上显示的彩色光斑就是某物。采样定理使这个方案成为可能。没有它，数字世界就不会存在。

So, here’s how Digital Light works: It extracts pixels from the real world, even from Mars or Venus—or from unreal worlds inside computers, as we’ll see—and sends them all over. These pixels carry only discrete position and intensity (or color) information. The actual light that seems to come from a pixel is only produced by a display device at the last possible moment before you see it with your eyes—not at any of those intermediate places around the world or among our sister planets. The smoothness of the original picture only reappears in the display. If the original creation of the pixels is done well, as per the Sampling Theorem, and if the reconstruction of those pixels into an analog display is done well, also per the Sampling Theorem, then what you see is an accurate representation of the original. We do get something from nothing. The pixel is the nothing—it has no dimension—and the displayed colored blob of light of the display is the something. The Sampling Theorem enables this scheme. The digital world wouldn’t exist without it.

间谍和扰频器

Spies and Scramblers

1936 年，当 Kotelnikov试图使其更广泛地传播时， Electrichestvo ( Electricity ) 期刊拒绝了 Kotelnikov 的论文——具有采样定理的论文。但那一年还有一件更耐人寻味的事情：科捷利尼科夫访问了美国！我对此有所暗示，并能够找到他下船记录的图像。上面显示他有60天的签证，他的旅行是由苏联政府支付的，他的目的地是纽约市的Amtorg贸易公司。25

Kotelnikov’s paper—the one with the Sampling Theorem—was rejected in 1936 by the Electrichestvo (Electricity) journal when he tried to get it into wider circulation. But there was a far more intriguing event that year: Kotelnikov visited the United States! I picked up a hint of this and was able to find an image of his disembarkation record. It reveals that he had a 60-day visa, his trip was paid for by the Soviet government, and his destination was the Amtorg Trading Corporation in New York City.25

Armand Hammer 于 1924 年成立了 Amtorg，以促进苏联和美国之间的贸易。他是一个迷人的人，是美国共产党创始人的儿子，他以美国社会主义工党的“手臂和锤子”标志命名。Armand 是一位著名的商人（西方石油公司的负责人）、慈善家和艺术品收藏家。

Armand Hammer had formed Amtorg in 1924 to facilitate trade between the USSR and the US. He was a fascinating man, son of a founder of the Communist Party USA, who named him after the “arm and hammer” logo of the Socialist Labor Party of America. Armand was a noted businessman (head of Occidental Petroleum), philanthropist, and art collector.

但 Amtorg 不仅仅是一家贸易公司。它是工业和军事间谍活动的温床。事实上，一份解密的 NSA（国家安全局）报告显示，美国在 1931 年花费了大量精力试图破解在 Amtorg 和莫斯科之间使用的密码。美国的努力失败了，因为“俄罗斯人使用一次性密码本进行加密”。26

But Amtorg was more than a trading company. It was a hotbed of industrial and military espionage. In fact, a declassified NSA (National Security Agency) report reveals that the United States expended major effort in 1931 trying to crack the codes that were used between Amtorg and Moscow. The US effort failed because “the Russians were using a one-time pad for their encipherment.”26

这一点特别有趣，因为依赖一次性预共享密钥的一次性密匙加密系统在接下来的十年里都没有被证明是有效的。Amtorg 间谍显然至少在没有证据的情况下信任它这么长时间。尽管我们并不确切知道 Kotelnikov 在 1936 年在那里做了什么，但创建和破解密码似乎比交易更有可能。

This is particularly interesting because the one-time pad encryption system, which depends on a one-time, pre-shared key, wasn’t proved effective for another decade. Amtorg spies obviously trusted it without proof for that long at least. Although we don’t know exactly what Kotelnikov was doing there in 1936, creating and breaking codes seems a lot more likely than trade.

或者他正在收集一次性垫使用情况的数据——因为证明一次性垫系统可靠性的人是 Kotelnikov。他在 1941 年 6 月 22 日希特勒入侵苏联前三天提交了他的证明。也许并不奇怪，香农也证明了一次性垫系统的可靠性；它于 1945 年出现在机密出版物中，并于 1949 年公开发表。27

Or perhaps he was gathering data on one-time pad usage—because the man who proved the reliability of the one-time pad system was Kotelnikov. He submitted his proof just three days before Hitler’s June 22, 1941, invasion of the Soviet Union. Perhaps not surprisingly, Shannon also proved the reliability of the one-time pad system; it appeared in a classified publication in 1945 and publicly in 1949.27

Kotelnikov 一定是在他的实验室里完成了他的一次性工作——这是斯大林秘密清洗之后唯一的 NKS 通讯委员会残余。该实验室幸存下来是因为军方迫切需要其无线电通信方面的机密工作。但随着德国人在 1941 年末迅速逼近莫斯科，科捷利尼科夫没有抓住他们可能突破的机会。他将他的实验室疏散到乌法，比喀山更东约 300 英里，莫斯科以东 800 英里。28

Kotelnikov must have done his one-pad work in his lab—the only remnant of the NKS communications commissariat after Stalin’s Secret Purge. The lab had survived because its classified work on radio communication was urgently needed by the military. But as the Germans fast approached Moscow in late 1941, Kotelnikov took no chances that they might break through. He evacuated his lab to Ufa, about 300 miles further east even than Kazan, so 800 miles east of Moscow.28

在斯大林格勒战役期间，俄罗斯前线一直依赖有线通信，并且在不可避免地出现电线故障时遭受了严重损失。Kotelnikov 的实验室生产了通过无线电而不是电线进行通信的安全设备，到 1943 年，战场上的军队开始使用这些设备。同年，他因其发展而获得了他的第一个斯大林奖。后来，在 1945 年 5 月，当局使用实验室的设备在德国签署投降协议时将莫斯科与苏联代表团联系起来。次年，科捷尔尼科夫获得了他的第二个斯大林奖。29

During the battle of Stalingrad, the Russian front had depended on wire communications and had suffered gravely when, inevitably, the wires failed. Kotelnikov’s lab produced secure devices that communicated via radio rather than wire, and by 1943 armies in the field were using the equipment. He received his first Stalin Prize that same year for its development. Then later, in May 1945, authorities used the lab’s equipment to connect Moscow to the Soviet delegation at the signing of Germany’s capitulation. Kotelnikov received his second Stalin Prize the following year.29

当实验室从乌法返回莫斯科时，它落入了内务人民委员会的控制之下。在这个危险时刻，莫斯科动力工程学院（MEI）新任院长戈卢布佐娃挺身而出，提出科捷尔尼科夫可能愿意回到她的机构，他的母校。他抓住了这个机会。她的干预将他作为绝密项目的负责人从 NKVD 的监狱系统中解救出来。30

When the lab returned to Moscow from Ufa, it fell under the control of the NKVD. At this dangerous moment Golubtsova, the new director of Moscow Power Engineering Institute (MEI), stepped forward and proposed that Kotelnikov might like to return to her institution, his alma mater. He leapt at this chance. Her intervention saved him, as head of a top-secret project, from the NKVD’s prison system.30

获得正确的数字表示

Getting Digital Representation Right

到目前为止，我所说的好像像素可以代表任何亮度级别。但是模拟曲线上的一个点是一个模拟值，这意味着它可以具有任何模拟无穷大的值。严格来说，图片亮度的模拟样本不是像素。它只有在转换为比特时才成为一个像素，然后它才会传播成为您在显示器上看到的光晕。一个数字样本，例如一个像素，只能采用某些离散值，它们的数量是有限的。

So far I’ve talked as if pixels could represent any brightness level whatsoever. But a point on an analog curve is an analog value, meaning it can have any of an analog infinity of values. Strictly speaking, an analog sample of a picture’s brightness isn’t a pixel. It only becomes a pixel when it’s converted to bits and only then will it be spread to become the glow you see on a display. A digital sample, such as a pixel, can take only certain discrete values, a finite number of them.

在我们在接下来的章节中讨论计算机之前，一些“计算机语言”在这里很方便。众所周知，位可以有两个值，通常称为 1 和 0。将位视为电灯开关。它有两个位置，通常称为上下。考虑两个电灯开关。他们可以担任多少职位？嗯，它们可以同时上升，或者都下降，或者以两种不同的方式一个上升另一个下降。所以答案是四个。换句话说，两个位（灯开关）可以有四个值。三可以有八个值。正如我们已经注意到的，第三个开关可以向上或向下，另外两个开关可以处于四个不同的位置。二乘四等于八。一般来说，随着电灯开关的数量增加一，开关可以采取的不同位置的数量增加一倍。为您省去麻烦，

Some “computerspeak” is handy here, before we get to computers in the next chapters. Famously a bit can have two values, usually called 1 and 0. Think of a bit as a light switch. It has two positions, often called up and down. Consider two light switches. How many positions can they be in? Well, they can both be up, or both down, or one up and the other down in two different ways. So the answer is four. In other words, two bits (light switches) can have four values. Three can have eight values. The third switch can be up or down, and the other two switches can be in four different positions, as we’ve already noticed. Two times four is eight. In general, as the number of light switches increases by one, the number of different positions the switches can assume doubles. To save you the trouble, I’ll simply tell you that 8 bits (light switches) can hold 256 values (positions), 10 bits can hold 1,024 values, and 16 bits can hold 65,536 values.

在计算机图形学的前几十年，通常将黑白图像的像素限制为 8 位。这意味着一个像素可以代表仅有的 256 种灰度中的一种。例如，值 0 和 255 可以分别代表黑色和白色，中间分布有 254 种其他灰色。但模拟曲线上的实际值可能是 49.673。要做什么？好吧，最接近的可用灰度值 50 将附加到该点的像素，因此当稍后将其重建到显示器上时，会在图片中引入一个小错误。亮度的微小舍入误差有多严重？50 是否“足够接近” 49.673 无关紧要？什么是像素亮度“足够接近”？

In the first decades of computer graphics it was common to limit a pixel of a black-and-white image to 8 bits. It meant that a pixel could represent one of only 256 shades of gray. For example, values 0 and 255 could represent black and white, respectively, with 254 other grays distributed in between. But the actual value on the analog curve might be 49.673. What’s to be done? Well, the closest available gray value, 50, would be attached to the pixel for that point, hence introducing a small error into the picture when it’s reconstructed later onto a display. How bad is that tiny round-off error in brightness? Is 50 “close enough” to 49.673 to not matter? What is “close enough” for pixel brightnesses?

在这种情况下，我们确实可以选择“足够接近”的值。医生，尤其是放射科医生，会在显示器上阅读 CAT 扫描和 MRI 扫描，并根据他们看到的情况做出诊断。因此，医疗行业的显示器制造商已经精确地测量了人类的灰度感知。他们提供的图表显示，普通人可以在我的特定桌面显示器上区分 630 种灰度。所以有两点很清楚。首先，过去只有 256 个可能值的 8 位像素还不够好。但是，其次，一个 10 位像素，有超过一千个可能的值就足够了。当您在一千多个选择中四舍五入到最接近的值时，人眼无法看到引入的错误。现代数字图像现在使用 16 位灰度像素，具有超过 65000 种灰度。这对于愚弄人脑来说已经绰绰有余了——任何人都可以进行任何展示。声音显示中的 16 位索像素也是如此。39

Here’s a case where we really can choose values that are “close enough.” Doctors, particularly radiologists, read CAT scans and MRI scans on displays and base their diagnoses on what they see. Display manufacturers for the medical profession, therefore, have measured human grayscale perception precisely. A chart they provide shows that normal humans can distinguish 630 shades of gray on my particular desktop display. So two things are clear. First, the 8-bit pixel of yore with only 256 possible values isn’t good enough. But, second, a 10-bit pixel, with over a thousand possible values is more than good enough. The human eye can’t see the error that’s introduced when you round off to the nearest value among a thousand-plus choices. Modern digital images now use 16 bits for a grayscale pixel, with over 65 thousand shades of gray. That’s easily more than enough to fool the human brain—any human for any display. The same goes for 16-bit soxels in the display of sound.39

您是否听过发烧友说 LP（或黑胶唱片）比 CD 更好？在视觉世界中，声称数码摄影永远无法与胶片摄影的微妙之处相提并论。这样的主张往往充满激情。他们有实质内容吗？嗯，当然有。有几种方法可能会导致错误地进行数字采样 - 每个像素或每像素的值不够、散布器糟糕或不够高采样频率。但这些主张的许多拥护者认为，他们批评的是数字化存在本质上的问题，而他们真正批评的是执行不力。如果一位工程师决定制作一张完全复制黑胶长播放 (LP) 专辑内容的 CD，那么理论上说她可以做到。同样，如果另一位工程师下定决心要从复制胶片照片的数字图像中打印出来，那么理论上说他可以做到。这两种情况都不容易。该理论必须尽可能接近理想地实践，但这是可能的。

Have you heard audiophiles say that LPs (or vinyl) are better than CDs? In the visual world the claim is that digital photography will never match the subtlety of film photography. Such claims are often held with great passion. Is there substance to them? Well, of course there is. There are several ways that digital sampling can be done incorrectly—not enough values per pixel or soxel, a lousy spreader, or insufficiently high sampling frequency. But many adherents of these claims believe they are criticizing something intrinsically wrong with digital, when what they’re really criticizing is poor execution. If an engineer set her mind on making a CD that replicated exactly what was on a vinyl long-playing (LP) album, the theory says that she could do it. Similarly, if another engineer set his mind to making a print from a digital image that duplicated a filmed photographic print, the theory says that he could do it. It wouldn’t be easy in either case. The theory would have to be practiced as near ideally as possible, but it would be possible.

一些发烧友声称能够区分以高于每秒 44,100 索像素 CD（光盘）标准的采样率录制的录音，该标准具有每索像素 16 位的特点。回想一下，人耳每秒可以听到高达 20,000 个周期（最高傅立叶频率），因此 Kotelnikov（或 Nyquist）采样率为每秒 40,000 索塞尔。CD 超过了 Kotelnikov 要求的从 soxels 进行完美模拟声音重建的最低要求。因此，如果该理论是可信的，那么没有人应该能够检测到使用更高采样率的系统对 CD 的改进。

Some audiophiles claim to be able to distinguish recordings made at higher sampling rates than the 44.1 thousand soxels-per-second CD (compact disc) standard which features 16 bits per soxel. Recall that the human ear can hear up to 20 thousand cycles per second—the highest Fourier frequency—so the Kotelnikov (or Nyquist) sampling rate is 40 thousand soxels per second. CDs exceed the minimum required by Kotelnikov for perfect analog sound reconstruction from soxels. So, if the theory is to be believed, then nobody should be able to detect an improvement over a CD with a system using a higher sampling rate.

专家可能能够检测到的另一种错误是舍入误差——索克的响度值数量不足。但正如已经提到的，没有人耳能做出比标准 CD 的 16 位索像素提供的更精细的区分——每个索像素大约有 65,000 个可能的响度级别。因此，对于每索像素使用更多位的系统，应该没有人能够检测到对 CD 的改进。

The other kind of error that an expert might be able to detect is round-off error—an insufficient number of values of loudness for a soxel. But as already mentioned, no human ear can make finer distinctions than those provided by the 16-bit soxel of the standard CD—with about 65,000 possible loudness levels per soxel. So nobody should be able to detect an improvement over a CD with a system using more bits per soxel.

两种在采样率和每像素位数方面都超过 CD 的标准数字音频系统是超级音频 CD (SACD) 和 DVD-Audio (DVD-A)。一项精心控制的科学测试得出了这样的结论：

Two standard digital audio systems that exceed the CD in both sampling rate and bits per soxel are Super Audio CD (SACD) and DVD-Audio (DVD-A). A carefully controlled scientific test came to this conclusion:

我们根据音乐类型和具体节目分析了所有的测试数据；高分辨率技术的类型；录音年龄；和听众年龄、性别、经验和听力带宽。这些变量都没有显示出与结果的任何相关性，或者答案和掷硬币结果之间的任何差异。40

We have analyzed all of the test data by type of music and specific program; type of high-resolution technology; age of recording; and listener age, gender, experience, and hearing bandwidth. None of these variables have shown any correlation with the results, or any difference between the answers and coin-flip results.40

换句话说，即使是发烧友也无法区分 CD 与 SACD 或 DVD-As。但是——这很能说明问题——同样的实验者报告说“几乎所有的 SACD 和 DVD-A 录音听起来都比大多数 CD 更好。” 在搜索与录音工程师的对话后，他们发现这是因为他们比制作典型 CD 录音的工程师更正确地使用了采样定理。教训是数字可以是模拟的准确表示，但必须正确完成。或者换一种说法，数字化并没有什么本质上的劣势。

In other words, even audiophiles couldn’t distinguish CDs from SACDs or DVD-As. But—and this is telling—the same experimenters reported that “virtually all of the SACD and DVD-A recordings sounded better than most CDs.” After searching conversations with the recording engineers, they found that this was because they used the Sampling Theorem much more correctly than the engineers who create the typical CD recording. The lesson is that digital can be an accurate representation of analog, but it has to be done correctly. Or stated another way, there is nothing intrinsically inferior about digital.

引领太空竞赛

Spearheading the Space Race

1957 年 8 月 27 日，科捷利尼科夫第二次访问美国，就在人造卫星发射之前——这一事件震惊了美国。这是国际地球物理年，标志着冷战时期科学交流中断的结束。科捷尔尼科夫现在是苏联科学院的正式成员，他正在为美国人携带来自俄罗斯的信息。他在科罗拉多州博尔德举行的一次会议上说，苏联将很快发射一颗卫星，该卫星将以每秒约 20 到 4000 万（兆）周期的频率进行广播。他被忽略了。美国人不会相信苏联科学已经取得了如此大的进步。但在 10 月 4 日，人造卫星以每秒 20.005 和 40.002 兆周期的速度发射和广播。太空竞赛如火如荼地进行着，科捷尔尼科夫也在场。42

Kotelnikov visited the United States a second time, on August 27, 1957, just before the launch of Sputnik—an event that shocked America to the bone. It was the International Geophysical Year, marking the end of the Cold War interruptions in scientific exchange. Kotelnikov was now a full member of the USSR Academy of Sciences, and he was carrying information from Russia for the Americans. He told a conference in Boulder, Colorado, that the Soviets would soon launch a satellite that would broadcast at frequencies of about 20 and 40 million (mega) cycles per second. He was ignored. Americans wouldn’t believe that Soviet science had progressed so far. But on October 4, Sputnik launched and broadcast at 20.005 and 40.002 megacycles per second. The space race was on in earnest, and Kotelnikov was there.42

差不多二十年后，当它结束时，他也在那里。1971 年在莫斯科举行的一次会议上，新任命的苏联科学院代理院长科捷尔尼科夫向美国外交官宣布，由于技术原因，拟议的阿波罗-礼炮试验任务将无法运行，但苏联已准备好继续取而代之的是重命名的阿波罗-联盟号测试项目。始于 1975 年 7 月的阿波罗-联盟号联合航天项目是两国缓和的最有力象征之一。43

He was there too when it ended, almost two decades later. At a 1971 meeting in Moscow, Kotelnikov, the newly appointed acting head of the USSR Academy of Sciences, announced to surprised US diplomats that the proposed Apollo-Salyut Test Mission wouldn’t work for technical reasons, but that the USSR was ready to pursue instead the renamed Apollo-Soyuz Test Project. The Apollo-Soyuz joint spaceflight project, which began in July 1975, was one of the most powerful symbols of the détente between the two countries.43

领导太空竞赛只是科捷尔尼科夫的成就之一。他完成了金星、水星、火星和木星的无线电定位，并因此获得了列宁奖。然后，配备 MEI 的太空任务金星 15 和 16 首次绘制了金星北部的地图——向整个太阳系发送像素。一颗名为 2726 Kotelnikov的小行星表彰了他对俄罗斯太空工作的诸多贡献。44

Spearheading the space race was just one of Kotelnikov’s achievements. He accomplished the radio location of the planets Venus, Mercury, Mars, and Jupiter and got a Lenin Prize for it. Then, MEI-equipped missions to space, Veneras 15 and 16, mapped the northern part of Venus for the first time—sending pixels across the solar system. An asteroid, 2726 Kotelnikov, honors his many contributions to the Russian space effort.44

摆脱高频

Getting Rid of High Frequencies

Kotelnikov 伟大的采样理念告诉我们将像素间隔“足够近”在一起意味着什么。如果违反了这条规则，就会发生坏事。您可能已经看到了结果：阶梯状边缘、马车车轮向后旋转、出现莫尔条纹条纹领带，或电子游戏中令人不快的背景闪光。这些文物是大数字融合早期的祸根。“是”，而不是“曾经”。我最近看了导演米开朗基罗安东尼奥尼的伟大电影l'Avventura和La Notte的 DVD 。在l'Avventura中，令人惊艳的 Monica Vitti 的波点连衣裙因采样错误而受损。她的裙子不规则地闪烁，仿佛波点是随机切换的小灯。几乎每一帧La Notte在被错误采样的窗户和建筑物的边缘闪闪发光。这样的数字伪影不仅是不必要的，而且是不可接受的。它们不是数字世界固有的，但它们是对采样定理的不知情使用的迹象。或者，在电子游戏的情况下，游戏计算机没有足够的能力来正确地进行采样。

Kotelnikov’s great sampling idea tells us what it means to have pixels spaced “closely enough” together. Bad things happen if this rule is violated. You’ve probably seen the results: stair-stepped edges, wagon wheels rotating backward, moiré patterns on striped neckties, or unpleasant background shimmer in videogames. These artifacts are the bane of the early days of the Great Digital Convergence. “Are,” not “were.” I recently watched DVDs of the director Michelangelo Antonioni’s great films l’Avventura and La Notte. In l’Avventura the stunning Monica Vitti’s polka-dot dress was marred because of incorrect sampling. Her dress blinked on and off irregularly as if the polka dots were little lights toggled randomly. And almost every frame of La Notte glittered at the edges of windows and buildings that were incorrectly sampled. Not only are such digital artifacts unnecessary, they’re unacceptable. They’re not intrinsic to the digital world, but they are signs of uninformed use of the Sampling Theorem. Or, in the case of videogames, of game computers without enough power to do sampling correctly.

如果您没有以足够高的速率进行采样，则会出现这些令人不快的伪影。采样定理说你必须以两倍最高傅里叶频率对视觉场景进行采样。因此，要么您必须以更高的速率进行采样以消除丑陋的伪影，要么您必须在采样之前消除​​场景中过高的频率。实用性通常决定后一种路径。

These unpleasant artifacts occur if you don’t sample at a high enough rate. The Sampling Theorem says you have to sample a visual scene at twice the highest Fourier frequency in it. So, either you have to sample at a higher rate to be rid of the ugly artifacts, or you have to get rid of the too-high frequencies in the scene before you sample. Practicality usually dictates the latter path.

频率峰值的直觉是锐利的边缘意味着高频。非常尖锐的边缘意味着非常高的频率。多高？无限高以获得完美锐利的边缘——也就是说，频率如此之高，你不可能处理它们。当然，您不能经常采样以准确表示它们。所以实用性再次介入。一般来说，要用像素表示一个场景，你首先必须摆脱它太高的频率，它太尖锐的边缘。窗口边缘就是很好的例子。有缺陷的La Notte DVD 的制作人显然没有遵循采样定理。他们在采样之前没有去除过高的频率，结果很丑陋——或者至少让人分​​心。

The frequencyspeak intuition is that sharp edges mean high frequencies. Very sharp edges mean very high frequencies. How high? Infinitely high for perfectly sharp edges—that is, frequencies so high you can’t possibly deal with them. Certainly you couldn’t sample often enough to accurately represent them. So practicality steps in again. In general, to represent a scene with pixels, you first have to get rid of its too high frequencies, its too sharp edges. Window edges are good examples. The producers of the flawed La Notte DVD clearly hadn’t followed the Sampling Theorem. They didn’t get rid of the too high frequencies before they sampled, and the result is ugly—or at least distracting.

有一种简单的方法可以从视觉场景中去除过高的频率。只需稍微散焦图片 - 但不要让您真正感觉到图片失焦。这种微妙的“涂抹”了所有的边缘。采样定理告诉你散焦多少就足够了。

There’s an easy way to remove too high frequencies from a visual scene. Simply defocus the picture slightly—but not so much that you can actually perceive that the picture is going out of focus. This subtlety “schmudges” out all the edges. The Sampling Theorem tells you just how much defocusing is enough.

围着暴君跳舞

Dancing around Tyrants

科捷尔尼科夫一生中最可耻的一幕——至少在许多西方人眼中——是 1975 年的安德烈·德米特里耶维奇·萨哈罗夫事件。萨哈罗夫是苏联氢弹之父之一，他已经放弃使用核武器，并积极争取不扩散核武器. 他和另一位持不同政见者索尔仁尼琴都受到苏联媒体的诽谤。但是萨哈罗夫获得了 1975 年的诺贝尔和平奖。作为回应，苏联科学院的成员发表了一份声明，谴责他和奖项。作为学院代理院长，科捷尔尼科夫向政治局提交了一份关于该事件的报告：

The most disgraceful episode in Kotelnikov’s life—at least to many Western eyes—was the Andrei Dmitrievich Sakharov affair of 1975. Sakharov, one of the fathers of the Soviet H-bomb, had renounced the use of nuclear weapons and campaigned actively for their nonproliferation. Both he and Solzhenitsyn, another dissident, were vilified in the Soviet media. But Sakharov received the Nobel Peace Prize for 1975. In response members of the USSR Academy of Sciences issued a statement denouncing him and the prize. As acting president of the Academy, Kotelnikov submitted a report about the event to the Politburo:

我们特此报告，该学院的 72 名成员签署了苏联科学家的声明，抗议将诺贝尔和平奖授予 AD Sakharov。45

We hereby report that seventy-two members of the Academy signed the statement of Soviet scientists protesting against the award of the Nobel Peace Prize to A. D. Sakharov.45

该报告还列出了五位拒绝签署声明的院士，其中包括另一位苏联氢弹之父维塔利·拉扎列维奇·金茨堡。作为回应，政治局禁止萨哈罗夫前往奥斯陆接受诺贝尔奖，1975年10月25日，科学家们签署的声明发表在苏联官方报纸《消息报》上。金茨堡写到科捷尔尼科夫在这件事中的角色：

The report also listed the five Academicians who refused to sign the statement, including Vitaly Lazarevich Ginzburg who was another of the fathers of the Soviet H-bomb. In response the Politburo forbade Sakharov to go to Oslo to receive the Nobel, and on October 25, 1975, the statement signed by the scientists was published in Izvestia, the official newspaper of the Soviet Union. Ginzburg wrote about Kotelnikov’s role in the affair:

我们的谈话相当平静。VA [Kotelnikov] 在恳求我，而不是威胁或恐吓我。总的来说，他是在毫无热情地完成任务，但那是他一贯的作风。我拒绝签字。

Our conversation was rather peaceful. V. A. [Kotelnikov] was pleading with me, instead of threatening or intimidating me. On the whole, he was carrying out the assignment without enthusiasm, but that was his usual manner. I refused to sign.

金茨堡继续说道，将当前时代（1991 年）与斯大林统治下的非常严酷的时代进行了对比：

Ginzburg continued, contrasting current times (1991) to the very harsh ones under Stalin:

我想我会在人身胁迫下签署这封信。. . [但是]显然没有被殴打或逮捕的危险，我仍然很困惑为什么这么多人签署了这样的信。46

I think I would have signed the letter under physical coercion . . . [but] there was obviously no danger of being beaten up or arrested, and it is still a mystery to me why so many people signed a letter of that kind.46

事实上，金茨堡和其他四人从未因拒绝而受到惩罚。Kotelnikov 和其他签名者是否还在遭受斯大林时代的恐惧？不管它的动机是什么，这封信只会增加对萨哈罗夫在西方持不同政见的支持，并使他成为英雄。47

Indeed Ginzburg and the other four were never punished for their refusal. Were Kotelnikov and the other signers still suffering from Stalinist-era fears? Whatever the motivations for it, the letter only increased support for Sakharov’s dissent in the West and made him a hero.47

从萨哈罗夫事件的角度来看，值得记住的是，1950 年代美国也发生过类似的事情。学者们转向美国原子弹之父 J. 罗伯特·奥本海默 (J. Robert Oppenheimer)，他同意萨哈罗夫对进一步使用核武器的看法。

To put the Sakharov affair in perspective, it’s worth remembering that something similar happened in America in the 1950s. Fellow academics turned on J. Robert Oppenheimer, father of America’s A-bomb, who shared Sakharov’s views on the further use of nuclear weapons.

像傅立叶一样，科捷尔尼科夫不得不巧妙地与他的暴君共舞——而且显然是这样做的。他在这方面非常成功，以至于在萨哈罗夫的信中，他除了领导科学院外，还担任着惊人的高位。因为科捷利尼科夫是俄罗斯最高苏维埃主席，苏联最大共和国的最高立法机构！八年。这怎么可能？

Kotelnikov, like Fourier, had to dance skillfully with his tyrants—and clearly did so. He was so successful at it that he held, at the time of the Sakharov letter, an astonishingly high position in addition to his leadership of the Academy of Sciences. For Kotelnikov was chairman of the Russian Supreme Soviet, the highest legislative body of the largest republic of the Soviet Union! For eight years. How could this be?

“一开始我很好奇，”他在30 年后告诉《消息报》记者。“系统是怎么运作的？然后我明白了。它尽可能简单。发言的人是中央亲自挑选的，发言稿是事先通过的。” 他补充说：“这一定是我一生中最轻松的工作。不过，最引人注目的。命运的讽刺。” 48

“I was curious at the beginning,” he told an Izvestia reporter 30 years later. “How does the system work? Then I got it. It was as simple as can be. Those who took the floor were handpicked by the Central Committee, draft speeches were approved beforehand.” He added, “It must have been the easiest job in my life. The most noticeable, though. An irony of fate.”48

也就是说，这是一个没有实权的职位。这是一个闲暇，本质上是他长期为国家服务的另一个奖项。

In other words, it was a position with no real power. It was a sinecure, essentially another award for his long service to his country.

平行与反讽

Parallels and Ironies

很明显，为什么今天“街上”的普通人不知道像素是什么。它需要了解采样定理，这需要了解傅里叶频率。公众通常不熟悉这些美丽而优雅的想法。然而，它们是数字媒体的基本理念——实际上，是一种通用的比特媒体。数字媒体主宰当今世界以及可预见的未来。本章和前一章试图传授对这个新世界的外行理解——对这两个伟大思想的直观理解。它们很容易总结：

It’s clear why a typical person “on the street” today doesn’t know what a pixel is. It requires a knowledge of the Sampling Theorem which requires a knowledge of Fourier’s frequencies. The public is generally unacquainted with either of these beautiful and elegant ideas. Yet they are the foundational ideas of digital media—actually, of the one universal medium of bits. Digital media dominate the world of today, and the foreseeable future. This and the preceding chapter attempt to impart a lay level of understanding of this new world—an intuitive understanding of the two great ideas. They are easy to summarize:

现实世界以一种明显连续的方式呈现给我们的感官。傅立叶的想法告诉我们，真实的连续世界可以描述为各种频率和幅度的波的总和——视觉世界可以被认为是我们眼睛的音乐。

The real world presents itself to our senses in an apparently continuous way. Fourier’s idea teaches us that the real continuous world can be described as a sum of waves of various frequencies and amplitudes—that the visual world can be thought of as music for our eyes.

采样定理教我们如何用离散样本来描述那些傅里叶波。因此，值得注意的是，视觉世界的模拟无穷大可以准确地编码为离散的、分离的、不可见的样本。在视觉世界中，这些样本称为像素。采样定理还告诉我们如何从这些离散样本中重建世界的连续表示：只需用散布器传播每个样本并将结果相加。在显示的时刻，每个离散像素在像素附近为重建的视觉场景贡献了一小块模拟无穷大。我们的眼睛理解这种重构的连续性，就像他们理解原始场景一样。

The Sampling Theorem teaches us how to describe those Fourier waves with discrete samples. Thus remarkably, the analog infinity of the visual world can be accurately encoded into discrete, separated, invisible samples. In the visual world those samples are called pixels. The Sampling Theorem also tells us how to reconstruct a continuous representation of the world from those discrete samples: simply spread each sample with a spreader and add up the results. At the moment of display each discrete pixel contributes a small dollop of analog infinity, in the near vicinity of the pixel, to a reconstructed visual scene. Our eyes make sense of this reconstituted continuity just as they would have made sense of the original scene.

这两个基本思想背后的人类戏剧与思想深刻一样引人入胜。傅立叶对拿破仑了解太多，因此几十年来，拿破仑将他从巴黎驱逐到外省。但效果是，傅立叶不受知识资本的喧嚣影响，创造了他的伟大想法，即世界是波浪的总和。Kotelnikov 被凶残的马林科夫的妻子 Valeriya Golubtsova 保护免于被关押在古拉格，在漫长而极其成功的职业生涯中继续发展他的采样定理的结果。傅立叶是在法国大革命中形成的；Kotelnikov 在俄罗斯的扩展革命中——从 1917 年十月革命到第二次世界大战和冷战。罗伯斯庇尔和拿破仑是傅立叶的暴君；斯大林、马林科夫、贝利亚和阿巴库莫夫至少间接属于科捷尔尼科夫。

The human drama behind these two essential ideas is as intriguing as the ideas are profound. Fourier knew too much about Napoleon, who consequently banished him from Paris to the provinces for decades. But the effect was that Fourier, shielded from the hurly-burly of the intellectual capital, created his great idea of the world as a sum of waves. And Kotelnikov, protected from incarceration in the Gulag by Valeriya Golubtsova, wife of the murderous Malenkov, continued throughout a long, immensely successful career to develop consequences of his Sampling Theorem. Fourier was forged in the French Revolution; Kotelnikov in Russia’s extended revolution—from the 1917 October Revolution, through World War II, and the Cold War. Robespierre and Napoleon were Fourier’s tyrants; Stalin, Malenkov, Beria, and Abakumov were Kotelnikov’s, indirectly at least. But Kotelnikov’s generic tyrant was really state security.

在像素的传记中，我们会反复发现收到的故事不一定是正确的。正确的往往比收到的要好——而且是真实的。Vladimir Kotelnikov，而不是美国的巨人克劳德香农，首先将采样定理带到了世界。

In the biography of the pixel we’ll repeatedly find that the received stories aren’t necessarily the right ones. And the right ones are often better than the received ones—in addition to being true. Vladimir Kotelnikov, not America’s giant Claude Shannon, first brought the Sampling Theorem to the world.

这两个人之间的相似之处是不可思议的。Kotelnikov 多年来一直担任苏联（今俄罗斯）科学院无线电工程与电子研究所所长，现在以他的名字命名。美国庞大的电子电气工程师协会 (IEEE) 最初是同名的无线电工程师协会。它将荣誉勋章授予香农，并在 2000 年将亚历山大·格雷厄姆·贝尔奖章授予科特尔尼科夫，标志着千禧年和大数字融合。具有讽刺意味的是，亚历山大·格雷厄姆·贝尔也与贝尔实验室同名——著名的香农的故乡。49

The parallels between the two men are uncanny. Kotelnikov was head of the Institute of Radio Engineering and Electronics of the USSR (today Russian) Academy of Sciences for many years, and it’s now named for him. America’s immense Institute of Electronic and Electrical Engineers (IEEE) was originally the similarly named Institute of Radio Engineers. It gave its Medal of Honor to Shannon, and it marked the millennium and the Great Digital Convergence by giving its Alexander Graham Bell Medal to Kotelnikov in 2000. The irony is that Alexander Graham Bell was also namesake to Bell Labs—famously Shannon’s home.49

尽管科捷尔尼科夫将采样定理带到了世界，但具有讽刺意味的是香农将它传授给了美国。如果不是因为香农从来没有为自己主张过采样定理，我们可能会援引“同时发明”的现象。我们将在下一章再次见到香农，国家安全将再次成为暴君——但这次是西方安全。

Although Kotelnikov brought the Sampling Theorem to the world, it’s a second irony that it was Shannon who taught it to America. We might invoke the phenomenon of “simultaneous invention” if it weren’t for the fact that Shannon never claimed the Sampling Theorem for himself. We’ll meet Shannon again in the next chapter, where state security will again be the tyrant—but Western security this time.

2003 年科捷尔尼科夫 95 岁生日——在他证明采样定理七十周年之际——在克里姆林宫的凯瑟琳大厅，普京总统授予他为服务勋章的全骑士（替代翻译：Chevalier、Knight）祖国，只有第四位持有该命令的人（图2.21）。Kotelnikov 于 2005 年 2 月 11 日去世。他只比香农多活了四年，但他们都看到了他们帮助定义的新千年和新时代。50

On Kotelnikov’s ninety-fifth birthday in 2003—on the seventieth anniversary of his proof of the Sampling Theorem—in the Catherine Hall of the Kremlin, President Putin made him a Full Cavalier (alternative translations: Chevalier, Knight) of the Order for Service to the Motherland, only the fourth person to hold the order (figure 2.21). Kotelnikov died on February 11, 2005. He outlived Shannon by only four years, but they both saw the new millennium and the new era that they helped define.50

下一章的主要目的是将数字光从现实世界扩展到虚幻世界。它的重点是数字光的第三个伟大的基础理念——计算。到目前为止，我讨论的像素都是从现实世界中派生出来的。例如，这包括科捷尔尼科夫从金星极地地区拍摄的那些像素，或者就此而言，我们每天用手机摄像头捕捉到的无数像素中的任何一个。另一方面，计算让我们创造了非常丰富但完全虚构的世界。它让我们从头开始生成或制作像素。当这些像素根据采样定理展开和添加并显示到我们的眼睛时，我们看到了一个不存在的世界。这怎么可能？

The main purpose of the next chapter is to extend Digital Light from the real world to unreal worlds. Its focus is the third great foundational idea of Digital Light—computation. The pixels I’ve discussed so far have been derived, taken, from the real world. That includes, for example, those pixels that Kotelnikov took from Venus’s polar regions, or, for that matter, any of the myriad pixels that we capture with our cellphone cameras daily. On the other hand, computation lets us create very rich, but completely fictitious worlds. It lets us generate, or make, pixels from scratch. When those pixels are spread and added per the Sampling Theorem and displayed to our eyes, we see a world that doesn’t exist. How is this possible?

圣图灵

St. Turing

他于 1912 年 6 月 23 日（一个世纪前）出生在伦敦，几乎正好在前一章的英雄科捷尔尼科夫和香农的出生之间。图灵在 22 岁时成为剑桥国王学院的研究员，在 34 岁时获得大英帝国勋章，在 38 岁时成为艾萨克·牛顿皇家学会会员，并在 6 月 7 日时死于 41 岁时的氰化物中毒， 1954. 所以——不像Kotelnikov 和 Shannon——他没有看到新千年和大数字融合。2

He was born in London on June 23, 1912—a century ago—almost exactly halfway between the births of Kotelnikov and Shannon, the heroes of the preceding chapter. Turing was a fellow at King’s College of Cambridge by age 22, received an Order of the British Empire by age 34, became a Fellow of Isaac Newton’s Royal Society by age 38, and was dead of cyanide poisoning by age 41, on June 7, 1954. So—unlike Kotelnikov and Shannon—he didn’t get to see the new millennium and the Great Digital Convergence.2

在 1960 年代，图灵对我们第一代计算机科学专业的学生来说是个谜。我们被教导了他的伟大想法——计算的概念——但他自己仍然是一个密码。从原本笼罩着他的迷雾中升起的是他已经自杀的持续谣言。

Turing was a mystery to us first-generation computer science students in the 1960s. We were taught his great idea—the notion of computation—but he himself remained a cipher. Rising out of the mist that otherwise obscured his person was the persistent rumor that he had committed suicide.

然后突然间，在 1983 年，安德鲁·霍奇斯出版了传记《艾伦·图灵：谜》，这最终使图灵这个人成为了人们关注的焦点。它解释了这个谜团：图灵帮助破解了纳粹德国的 Enigma 密码并赢得了战争。但他在布莱切利公园的工作是绝密的，并被英国的官方保密法扣押了数十年。3

Then suddenly, in 1983, Andrew Hodges published the biography Alan Turing: The Enigma, which finally brought Turing the man into sharp focus. It explained the mystery: Turing had helped crack Nazi Germany’s Enigma code and win the war. But his work at Bletchley Park was top secret and had been impounded for decades by Britain’s Official Secrets Act.3

超越这一启示是另一回事。当同性恋仍然是一种犯罪行为时，图灵就已经是公开的同性恋了，甚至是鲁莽地这样做了。当他在1952年因“不雅行为”被捕时，他无法利用他拯救了英格兰的事实来拯救自己。不雅行为胜过《保密法》。考虑到监狱或化学阉割的选择，他选择了阉割。他的马拉松运动员的身体（图 3.1）从荷尔蒙和发达的乳房。也许，正是这种屈辱驱使他吃了一个（可能）毒苹果——在一个本可以直接从迪斯尼白雪公主中取出的死亡场景中。4

Topping that revelation was another. Turing had been openly gay, even recklessly so, when homosexuality was still a crime. When he was arrested for “indecent acts” in 1952, he couldn’t use the fact that he had saved England to save himself. The indecent acts trumped the Secrets Act. Given the choice of prison or chemical castration, he chose castration. His marathon runner’s body (figure 3.1) grew fatter from the hormones and developed breasts. It was the humiliation, perhaps, that drove him to eat a (possibly) poisoned apple—in a death scene that could have been lifted straight out of Disney’s Snow White.4

传记作家霍奇斯本人是国王学院的理论物理学家，也是英国同性恋解放运动的成员，他终于揭开了神秘的面纱。他得到了完整的故事，并仔细、亲密、很好地讲述了它。5

Biographer Hodges, himself a King’s College theoretical physicist and a member of Britain’s gay liberation movement, finally parted the veil of secrecy. He got the full story and told it carefully, intimately, and well.5

在 1950 年代的美国，在参议员约瑟夫麦卡锡的恐怖统治下的“薰衣草恐慌”中，共产党人和“性变态者”被归为国家的敌人。英格兰也遭受了同样的错觉，尤其是在 1951 年以及臭名昭著的受过剑桥教育的间谍和同性恋者盖伊·伯吉斯 (Guy Burgess) 叛逃到莫斯科之后。图灵对共产主义从来没有任何兴趣——他对此嗤之以鼻——而且他是如此公开坦率的同性恋，以至于他不可能被勒索。但麦卡锡和军情五处没有做出这些区分，而图灵有共产主义朋友。6

In 1950s America, Communists and “sexual perverts” were lumped together as enemies of the state in the “lavender scare” of Senator Joseph McCarthy’s reign of terror. England suffered the same delusion, especially after 1951 and the infamous defection to Moscow of Cambridge-educated Guy Burgess, a spy and a homosexual. Turing never had any interest in Communism—he mocked it—and he was so openly and candidly gay that he couldn’t have been blackmailed. But McCarthy and MI5 didn’t make these distinctions, and Turing had Communist friends.6

剑桥大学著名的秘密社团 Apostles 将 Burgess 列为其成员之一。他们中的许多人是马克思主义者或同性恋者或两者兼而有之。图灵的两个最好的朋友和合作者罗宾·甘迪和大卫·尚佩诺恩也是使徒，这对图灵的事业没有帮助。尤其是甘迪，在学生时代是共产党员。

The Apostles, a prestigious secret society at Cambridge, counted Burgess among its members. Many of them were Marxists or gay or both. It can’t have helped Turing’s cause that two of his best friends and collaborators, Robin Gandy and David Champernowne, were also Apostles. Gandy, in particular, was a member of the Communist Party during his student days.

每个人都同意图灵死于氰化物中毒，但他是如何中毒的仍然存在争议。毒苹果自杀是最流行的理论。但实际的苹果从未进行过氰化物测试，为其他解释敞开了大门。他的母亲认为这是一场意外。但另一种说法是政府安全部队暗杀了他。7

Everybody agrees that Turing died of cyanide poisoning, but just how he was poisoned is still controversial. The poison-laced-apple suicide is the most popular theory. But the actual apple was never tested for cyanide, leaving the door open for alternative explanations. His mother believed it was an accident. But another theory is that government security forces assassinated him.7

具有讽刺意味的是，在图灵毒尸被发现后的第二天，麦卡锡的反共恐怖统治终于落下帷幕。1954 年 6 月 9 日那天，美国陆军首席法律顾问约瑟夫·韦尔奇斥责这位参议员，因为他试图暗杀另一个被指控为共产主义的人。

It’s a sad irony that the very day after the discovery of Turing’s poisoned body, McCarthy’s anti-Communist reign of terror was finally brought to heel. On that day, June 9, 1954, Joseph Welch, head counsel for the United States Army, famously rebuked the senator as he attempted to assassinate the character of yet another person with accusations of Communism.

英国政府最终在 1967 年放宽了虚伪的反同性恋法律，并在 2009 年为其骇人听闻的错误公开道歉。但这两个事件都来得太晚了，无法阻止“圣。图灵。” 2012 年全球庆祝他诞辰 100 周年为他辩护——最终在 2013 年圣诞节前夕女王的死后赦免结束。8

The British government eventually relaxed its hypocritical anti-gay laws in 1967 and apologized publicly in 2009 for its appalling mistake. But both events came way too late to forestall the martyrdom of “St. Turing.” The 2012 worldwide celebrations of the centenary of his birth were his vindication—capped off finally by the Queen’s posthumous pardon on Christmas Eve, 2013.8

傅立叶的法国恐怖和科捷尔尼科夫的大恐怖都极其致命。麦卡锡的恐怖，以及相应的英格兰环境，不在同一个联盟中，但它们仍然毁掉了成千上万的生命——并直接导致了图灵的死亡。图灵的暴君不是皇帝也不是党主席，而是国家安全。他没有在肉体上被监禁，但他被《官方保密法》在精神上囚禁——在一个隐喻的布莱切利公园 sharashka 中。

Fourier’s French Terror and Kotelnikov’s Great Terror were vastly and horribly deadly. McCarthy’s terror, and the corresponding milieu in England, weren’t in the same league, but they ruined thousands of lives nevertheless—and led directly to Turing’s death. Turing’s tyrant wasn’t an emperor or a party chairman, but state security. He wasn’t physically imprisoned, but he was held mentally captive by the Official Secrets Act—in a metaphorical Bletchley Park sharashka.

在图灵的故事中，我们再次找到了技术突破的要素：他的伟大计算思想；第二次世界大战的混乱和恐惧推动了计算机的发展，这些机器实现了他的想法；以及国家机密的暴政，将科学家集中在一个地方，并以一种不正当的方式保护他们。

In Turing’s story we again find the ingredients for technological breakthrough: his great idea of computation; the chaos and fear of World War II, which drove the development of computers, the machines that realized his idea; and the tyranny of state secrecy that concentrated the scientists in one place and protected them in a perverse way.

延展性

Malleability

计算机在每个步骤中所做的事情都是微不足道的。一个典型的步骤是这样的：从这里取一些比特，稍微旋转一下，然后把得到的比特放在那里。旋转可能只不过是将每个 0 换成 1，反之亦然，或者每个 0 或 1 向右移动一个位置。

What a computer does at each step is trivial. A typical step has this form: take some bits from here, twiddle them a little, and put the resulting bits there. A twiddle might be nothing more than a swap of each 0 for a 1 and vice versa, or a move of each 0 or 1 a single position to the right.

这些简单可笑的步骤串在一起的长序列使计算机变得强大。这并不明显——无意识的步骤应该导致有意识的结果。那么愚蠢的序列如何变得有意义呢？魔法在哪里？

Long sequences of these laughingly simple steps strung together are what make the computer powerful. It’s not obvious that this should be so—that mindless steps should lead to mindful results. So how does a dumb sequence become meaningful? Where does the magic lie?

随机序列没有什么有趣的。但是一个有意义的序列会导致计算机执行一个有用或有趣的过程。您知道的任何计算机或手机应用程序都是由一系列简单步骤执行的过程。另一个序列产生了皮克斯电影。

A random sequence does nothing interesting. But a meaningful sequence causes a computer to implement a process that’s useful or entertaining. Any computer or phone app you know is such a process performed by a sequence of those simple steps. Another sequence results in a Pixar movie.

计算机硬件所做的很简单——那些微不足道的步骤。但要创建这些步骤的有意义的序列，需要相当大的智力。那就是软件设计。具有这种能力的有创造力的人是程序员。所以，魔法就在他们的脑海里。

What computer hardware does is simple—those trivial steps. But to create meaningful sequences of those steps requires considerable mental prowess. That’s software design. The creative humans who are endowed with such prowess are programmers. So, the magic lies in their minds.

这就像一架音乐会三角钢琴。钢琴可以演奏无数的音调。那是执行简单步骤的硬件。大多数序列只是噪音。但偶尔会出现一些产生美妙音乐的软件。由程序员肖邦诱导的音符序列变成了练习曲、华尔兹或马祖卡。

It’s like a concert grand piano. The piano can play countless sequences of tones. That’s hardware executing simple steps. Most of the sequences are just noise. But occasionally some software appears that produces glorious music. The sequence of notes induced by programmer Chopin becomes an étude, a waltz, or a mazurka.

一台计算机，就像一架钢琴，可以实现无数简单的基本步骤序列。因此，它可以实现无数复杂而有意义的流程它们——可以说是它的音乐——只使用 2 位而不是 88 个钢琴键。正是“无数”这个词捕捉了延展性的超越性，这是一种对计算机至关重要的品质，以至于它应该获得“大写 M”的地位。我们人类的思想被计算机可以实现的广泛的可能过程所淹没。总是有另一个有意义的无意义步骤序列来产生我们想要使用的东西。总有另一个应用程序、游戏或电影要构建。而一个工具，计算机，可以完成所有这些。

A computer, like a piano, can implement countless sequences of its simple basic steps. It can therefore implement countless complex and meaningful processes with them—its music, so to speak—using only 2 bits rather than 88 piano keys. It’s the word “countless” that captures the transcendence of Malleability, a quality so crucial to computers that it deserves its “capital M” status. Our human minds are swamped by the sheer breadth of possible processes that a computer can realize. There’s always another meaningful sequence of meaningless steps to produce something we want to use. There’s always another app, game, or movie to build. And one tool, the computer, does them all.

事实上，延展性本质上是计算的一部分，正如我们将在本章中看到的那样。这是概念的礼物。但延展性的荣耀伴随着一个谜团——不可知性的谜团。尽管计算机采取的每一步都是完全可预测的，但我们并不总是知道这些步骤的顺序将如何展开。在小处确定并不意味着在大处确定。必须确定一个计算——必须有一些关于一系列完全确定的步骤将做什么的答案——但我们并不总是知道那个答案。确定和预定之间的细微差别是潜伏在计算核心的奥秘。因此，计算机简单而深刻。

In fact, Malleability is fundamentally part of what computation means, as we’ll see in this chapter. It’s a gift of the concept. But the glory of Malleability comes with a mystery attached—the mystery of unknowability. Although each step that a computer takes is completely predictable, we can’t always know how a sequence of those steps will unfold. Determined in the small doesn’t mean predetermined in the large. A computation must be determined—there must be some answer about what a sequence of completely deterministic steps will do—but we can’t always know that answer. The subtle difference between determined and predetermined is the mystery that lurks at the heart of computation. So a computer is simple but profound.

放大

Amplification

另一种非常不同的超越是放大。很明显，计算机可以重复一个动作——它的任何无意识步骤的序列。这很简单。重要的是它可以以我们无法理解的速度重复这个动作无数次。不是无数次。我们可以计算重复次数——或者至少我们的计算机可以。我们只是无法理解与我们人类思维有关的数字。淹没我们的是计算机进程的规模。11-11 skydillion 是一个虚构的愚蠢数字——斯克罗吉叔叔遇到比尔博巴金斯——但它让人想起计算中的数字是多么愚蠢，即使它们是实际数字。我们无法想象它们。正如斯托帕德的瓦尔在题词中所说，“你必须疯了”才能对付它们。精神错乱可能还不够。10

The other, very different kind of transcendence is Amplification. It’s obvious that a computer can repeat an act—any sequence of its mindless steps. That’s simple. What matters is that it can repeat the act an impossible number of times at speeds we can’t understand. Not a countless number of times. We can count the repetitions—or at least our computers can. We just can’t fathom the numbers involved with our human minds. It’s the scale of computer processes that swamps us. Eleventy-eleven skydillion is an invented silly number—Uncle Scrooge meets Bilbo Baggins—but it evokes how silly the numbers in computations are, even when they are actual numbers. We can’t imagine them. As Stoppard’s Val says in the epigraph, “you’d have to be insane” to deal with them. And insanity probably wouldn’t suffice.10

重复您已经对计算机进行编程的任务是微不足道的。只需指示计算机根据需要多次重复该任务。几乎不需要额外的智力努力。但是，与这种简单性相反的是加速计算机以达到难以想象的数字所需的智力。这就是硬件设计。具有这种能力的创造性人类是工程师。

Repeating a task that you’ve already programmed a computer to do is trivial. Simply instruct the computer to repeat the task as many times as you require. There’s almost no additional intellectual effort required. But, countering that simplicity is the mental prowess that’s required to speed up the computer to reach the unimaginable numbers. That’s hardware design. The creative humans with that prowess are engineers.

在电气工程中，放大意味着增加很多。指数是衡量“很多”的方式——例如，不仅仅是两倍或三倍，而是平方或立方。增加不是 2 倍或 3 倍，而是 2 的指数或 3 的指数好几倍——不仅仅是 2 x或 3 x，而是x ²或x ³。如果输入信号是 10 伏，那么放大后的输出不仅仅是 20 或 30 伏，而是 100 或 1,000 伏。放大是技术的礼物。最早的计算机的建造正是因为它们可以超越人类——它们可以增强人类的能力。

In electrical engineering to amplify means to increase—by a lot. Exponentially is how “a lot” is measured—not mere doubling or trebling, for example, but squaring or cubing. The increase is not 2 times or 3 times better but exponent-of-2 or exponent-of-3 times better—not merely 2x or 3x, but x² or x³. If an input signal is, say, 10 volts, then an amplified output is not just 20 or 30 volts, it’s 100 or 1,000 volts. Amplification is a gift of technology. The earliest computers were built precisely because they could outstrip human beings—they could amplify human capabilities.

第一台计算机将人类能力提高了 10,000 倍。到 1965 年，这一因素已跃升至 1,000,000——至少——是通过制造更大的机器并将基础技术从真空管转换为晶体管而实现的。我们将其称为 Epoch 1 加速。

The very first computer amplified human capability by 10,000. By 1965 that factor had jumped to 1,000,000—at least—accomplished by building ever larger machines and by converting the underlying technology from vacuum tubes to transistors. We’ll call this the Epoch 1 speedup.

但超越放大——放大本身的指数放大——伴随着更现代的技术——集成电路：大量晶体管和它们之间的导线在一个芯片上。1965 年，现在被称为摩尔定律的声明是开启下一个显着加速的绿旗——第二纪元。人类的能力不仅被计算机以指数方式放大，而且，这种放大的速度也以指数方式增加。

But transcendent Amplification—exponential amplification of amplification itself—came with even more modern technology, the integrated circuit: lots of transistors and the wires between them on a single chip. In 1965 the statement now known as Moore’s Law was the green flag that started the next dramatic speedup—Epoch 2. Human capabilities were not only amplified exponentially by computers, but, dramatically, the rate of that amplification increased exponentially too.

简而言之，摩尔定律暗示放大率将每五年上升到 10 的下一个指数——下一个 10 的因子。因此，如果人类的扩增在 1965 年是 10 ⁶（100 万）——事实上——那么它在 1970 年将是 10 ^{7 （1000 万），在 1975 年是 10 8}（1 亿），以此类推。它会让早期计算机的创造者感到惊讶，就像今天让我们感到惊讶一样。另一个惊喜是 Epoch 2 中更强大的计算机变得更小，而不是更大。

In a nutshell Moore’s Law implied that Amplification would go up to the next exponent of 10—the next factor of 10—every five years. So if Amplification of humans was by 10⁶ (1 million) in year 1965—as it was—then it would be 10⁷ (10 million) in 1970, and 10⁸ (100 million) in 1975, and so forth. It would have amazed the creators of the early computers as much as it amazes us today. Another surprise was that the more powerful computers of Epoch 2 got smaller, not larger.

就其本质而言，我们无法理解摩尔定律的含义。这是一个数量级的事情。我们人类通常无法看到超过 10 倍的变化。我们在那里碰上了一堵概念墙。所以我们用大的表达数量级来暗示僵局而不是平淡无奇，简单的算术因子10. 宏大的表达意味着一个如此巨大的变化，以至于只有新的思维过程——新的概念化——才能处理它。它不仅仅是更多——它是不同的。我们必须先掌握一个数量级的改进，然后才能开始思考下一个订单可能意味着什么。然而，计算机已经将人类的放大率提高了至少 17 个数量级！前六个订单（一百万倍）出现在摩尔定律之前，之后又有 11 个订单（一千亿倍）。总而言之，有一百亿倍的放大——超出任何人最疯狂的想象，无论是否天才。

By its very nature we can’t understand the implications of Moore’s Law. It’s an order-of-magnitude thing. We humans can’t usually see beyond a factor-of-10 change. We hit a conceptual wall there. So we use the grand expression order of magnitude to connote the impasse rather than the bland, simply arithmetical factor of 10. The grand expression implies a change so great that only new thought processes—new conceptualizations—can handle it. It’s not simply more—it’s different. We have to master one order-of-magnitude improvement before we can even start to contemplate what the next order might mean. Yet computers have improved the Amplification of humans by at least 17 orders of magnitude! The first six orders (a millionfold) came before Moore’s Law, and 11 more have come since (a hundred-billionfold). All told there’s been a hundred-quadrillionfold Amplification—beyond anyone’s wildest imaginings, genius or not.

计算机利用我们人类的智慧超越了我们固有的数量级限制。它将我们微不足道的人类行为放大到我们没有帮助的物种无法达到和无法想象的极端。增幅的荣耀伴随着那些数量级障碍的奥秘。

A computer leverages our human intellect past our inherent order-of-magnitude limitations. It Amplifies our puny human-sized actions to extremes unreachable—and unimaginable—by our unaided species. The glory of Amplification comes with the mystery of those order-of-magnitude barriers.

拍摄或制作，拍摄或计算

Take or Make, Shoot or Compute

Kotelnikov 的采样定理告诉我们，相机可以将现实世界变成像素。换句话说，我们可以用相机从真实的视觉世界中获取像素——我们可以拍摄它们。然后我们可以使用它们来重建那个视觉世界的样子——即使是在更晚的时间或很远的地方。

Kotelnikov’s Sampling Theorem taught us that a camera can turn the real world into pixels. In other words, we can take pixels from the real visual world with a camera—we can shoot them. Then we can use them to reconstruct what that visual world looked like—even at a much later time or far distant place.

或者我们可以从头制作像素——我们可以计算它们。如果我们显示的像素按照与现实世界像素相同的规则运行，那么我们也可以看到虚幻世界。新千年的完整数字之光源于这种想法。

Or we can make pixels de novo—we can compute them. If we display pixels that play by the same rules as real-world pixels, then we can see unreal worlds too. The full Digital Light of the new millennium flows from that thought.

我们利用延展性和放大功能为我们提供了成熟的数字光——例如，一部完全数字化的故事片，或者最热门的视频游戏，或者对国际互联网体验的即时视觉访问。计算机创建像素，并从视觉世界中获取它们。

We’ve harnessed Malleability and Amplification to give us full-blown Digital Light—a completely digital feature film, for example, or the hottest videogame, or instantaneous visual access to the international internet experience. Computers create pixels, as well as taking them from the visual world.

本章是关于延展性的超越性，这是艾伦·图灵发明计算的礼物。这是关于想法的。下一章是关于速度的。它是关于硬件计算机的发明，以实现计算的想法。这是关于放大的超越，技术的礼物。这也是关于数字光的黎明——令人惊讶的巧合——与计算机同时发生。通俗地说，本章和下一章是关于摩尔定律到来之前理论和实践——科学和工程——的英雄。在后面的章节中，当摩尔定律出现并在 Epoch 2 加速中驱动放大超新星时，我们重新审视“定律”并更仔细地检查它。

This chapter is about the transcendence that is Malleability, the gift of Alan Turing’s invention of computation. It’s about ideas. The next chapter is about speed. It’s about the invention of hardware computers to realize the idea of computation. It’s about the transcendence that is Amplification, the gift of technology. And it’s also about the dawn of Digital Light which—by a surprising coincidence—occurred simultaneously with that of computers. In mundane terms this chapter and the next are about the heroes of theory and practice—of science and engineering—before the arrival of Moore’s Law. In a later chapter, when Moore’s Law arrives and drives Amplification supernova in the Epoch 2 speedup, we revisit “the Law” and inspect it more closely.

他有算法

He’s Got Algorithm

图灵的重大突破始于系统过程的想法。这是一个仔细或系统地做某事意味着什么的例子。假设您家中的一位客人，您所在城镇的陌生人，想知道如何去杂货店。你的步行方向可能是这样的：走出前门，在街上右转。步行到第一个十字路口，然后再往前走两个街区。在那里的街道左转，沿着它走三个街区。继续穿过那个十字路口，商店就在右侧，从十字路口进去的第四栋建筑。

Turing’s great breakthrough starts with the idea of a systematic process. Here’s an example of what it means to do something carefully or systematically. Suppose a guest in your home, a stranger to your town, wants to know how to get to a grocery store. Your walking directions might be something like this: Go out the front door and turn right at the street. Walk to the first intersection and then go two blocks beyond. Turn left at the street there and walk along it three blocks. Proceed through that intersection, and the store is on the right side, the fourth building in from the intersection.

因此，直觉上，谨慎或系统意味着进入商店的较大过程被分解为一系列较小的步骤，其中每个步骤或说明对大多数人来说都是简单、明确且显而易见的。一位客人按照给定的顺序准确地执行每一项，一定会到达商店。

Intuitively, then, being careful or systematic means that the larger process of getting to the store is broken down into a sequence of smaller steps, where each step or instruction is simple, unambiguous, and obvious to most people. A guest who accurately performs each, in the order given, will surely arrive at the store.

上面的步行路线是一个简单的说明列表。客人从列表的一端开始，在另一端结束。要遵循的指令数等于列表中的步骤数。然而，一般来说，系统化过程的结构比这更多。考虑一下将钉子钉入木板的方法：（1）取钉子。(2) 如果没有，那么你就完成了，否则继续。(3) 敲钉子。(4) 如果指甲弯曲，则将其拉直并重复步骤 3，否则从步骤 1 重新开始。

The walking directions above are a simple list of instructions. The guest begins at one end of the list and finishes at the other. The number of instructions to be followed equals the number of steps in the list. Generally, however, there’s more structure to a systematic process than that. Consider this one for hammering nails into a board: (1) Fetch a nail. (2) If there isn’t one, then you’re done, otherwise proceed. (3) Hammer in the nail. (4) If the nail bends, then straighten it out and repeat step 3, otherwise start again at step 1.

此列表与步行路线具有相同数量的说明，但在强大的方面有所不同。它有循环。只要有钉子，您就重复外环（步骤 1 到 4），只要钉子在不恰当地敲打时弯曲，就重复内部“嵌套”环（步骤 3 和 4）。你在这个列表的一端进入，你可能永远不会出来——比如说，如果你每次敲钉子时都会弯曲一个钉子。执行的步骤数通常比指令数大得多——有时甚至大得多。

This list has the same number of instructions as the walking directions, but it’s different in a powerful way. It has loops. You repeat the outer loop (steps 1 through 4) so long as there are nails, and the inner “nested” loop (steps 3 and 4) so long as a nail gets bent when it’s inexpertly hammered. You enter at one end of this list and you might never come out—if you bend a nail every time you hammer one, say. The number of steps executed is usually much larger—sometimes vastly larger—than the number of instructions.

钉钉方向有两个分支步骤，形式为“如果 . . . 然后 。. . 否则 。. 。” （步骤 2 和 4）。这些被称为条件分支。如果满足某个条件，则分支到某某步，否则分支到某某步。条件分支（由if表示）允许您更改执行列表中指令的顺序。

There are two branching steps in the nailing directions that take the form “if . . . then . . . otherwise . . .” (steps 2 and 4). These are called conditional branches. If some condition is satisfied, then branch off to step so-and-so otherwise branch off to step such-and-such. A conditional branch—signified by an if—lets you change the order in which you perform the instructions in a list.

它还可以让您逃脱无限循环。例如，考虑这个：（1）说“你好”。(2) 转到第 1 步。没有条件分支可以用来转义循环。因此，一旦这个系统过程开始，它就永远不会停止。

It also lets you escape an infinite loop. Consider this one, for example: (1) Say “Hello.” (2) Go to step 1. There’s no conditional branch with which to escape the loop. So once this systematic process starts, it never halts.

条件分支——系统过程的if ——是计算的关键。可以执行条件分支的机器比不能执行的机器强大得多。它可以简洁地导致一个难以想象的长度的过程小心地展开。它可以实现自我循环的流程——也许是十一十一​​天——甚至可以修改自己作为流程的一部分。任何声称是计算机的机器都必须具有条件分支指令。一台机器可以有十层楼高，全电子化，速度极快，但如果它没有条件分支指令，那么它就不是计算机。

Conditional branches—the ifs of systematic process—are key to computation. A machine that can execute conditional branches is much more powerful than one that cannot. It can succinctly cause a process of unimaginable length to unfold carefully. It can implement processes that loop on themselves—perhaps eleventy-eleven skydillion times—or even modify themselves as part of the process. Any machine that purports to be a computer must have a conditional branch instruction. A machine can be ten stories tall, fully electronic, and extremely fast, but if it doesn’t have a conditional branch instruction, then it’s not a computer.

如果它只计算数字，那么它就不是计算机。系统过程的概念远大于数字。即使是我们第一个系统过程的例子，步行和锤击指令，也不是关于数字的。毫不奇怪，图灵很早就完全理解了这个概念。

And if it only calculates numbers then it’s not a computer. The notion of systematic process is much larger than numbers. Even our first examples of systematic process, the walking and hammering instructions, weren’t about numbers. Not surprisingly, Turing understood this notion early and completely.

并不是说数字不重要。它们是我们几千年来一直系统化的事情之一。即使在这个计算器无处不在的时代，孩子们仍然学习如何将两个十进制数字相加。众所周知，这样做的方法是将每对数字从右边开始依次相加，然后将它们和的一个数字写入答案。如果（就是这样！）这些数字的总和是 10 或更大，你将“1”带到下一步，向左一个位置，并将其添加到那里。等等。在这个过程中还有另一个隐含的情况：如果你用完了数字或携带 1 来添加，则停止。这是通常称为加法算法的过程的草图. 我敢肯定，我们都很高兴我们的计算器现在为我们“记住”了这个算法的细节，并且对它们足够了解，如果我们坚持的话，它们可以添加两打数字，而不仅仅是两个。

Not that numbers aren’t important. They’re one of the things we’ve been systematic about for millennia. Even in this era of the omnipresent calculator, kids still learn how to add two decimal numbers together. Famously, the way to do it is to add each pair of digits successively, starting at the right, and write one digit of their sum to the answer. If (there it is!) the sum of those digits is 10 or larger, you “carry the 1” to the next step, one position to the left, and add it in there. And so forth. There’s another if implicit in this process: if you run out of digits or carried 1s to add, then halt. This is a sketch of the process that’s often called the addition algorithm. We’re all happy, I’m sure, that our calculators now “remember” the details of this algorithm for us and know them well enough that they can add two dozen numbers, if we insist, rather than just two.

在上个世纪，算法这个词成为我们一直称之为“系统过程”的同义词。这个词是对 9 世纪波斯算术专家花拉子米名字的诬告。他在古代巴格达写了关于涉及十进制数系统的系统过程 - 一个来自印度的新概念，其中包含一个称为 0 的奇怪新事物。后来，在中世纪晚期，印度数字系统本身被称为algorism或augrym，还有两个以他的名字命名的腐败。因此，算法的概念植根于数字和数字操作——但绝不受其约束。

In the last century the word algorithm became synonymous with what we’ve been calling “systematic process.” The word is a corruption of the name of a ninth-century Persian arithmetic expert, al-Khwarizmi. He wrote in ancient Baghdad about systematic processes involving the decimal number system—a new concept just in from India featuring a strange new thing called 0. Later, in the late Middle Ages, that Indian number system itself came to be called algorism or augrym, two more corruptions honoring his name. So the notion of an algorithm is rooted in numbers and number manipulation—but by no means bound by it.

十四世纪的杰弗里乔叟使用奥格利姆中的名词来描述星盘上的标记。但他的朋友 Thomas Usk 以更令人印象深刻的方式使用了这个词。他的题词是关于 0 的力量——在augrym 中的 sypher。它本质上是说，虽然单独的 0 什么都不是，但附加的 0 很多。用我们的话来说，每个附加的 0 都是一个数量级的增加——确实是很多东西。他的预兆远不及我们的算法，但事后看来，他的格言似乎是对放大潜力的早熟观察。12

Geoffrey Chaucer in the fourteenth century used nombres in Augrym to describe markings on an astrolabe. But his friend Thomas Usk used the term in a more impressive way. His epigraph is about the power of the 0—the sypher in augrym. It essentially says that while a 0 alone is nothing, a 0 appended is a lot. In our terms, each appended 0 is an order-of-magnitude increase—a lot of something indeed. His augrym was nowhere near our algorithm, but his aphorism appears in hindsight to be a precocious observation about the potential of Amplification.12

当算法中的步骤数变大、循环数成倍增加、嵌套层次加深、条件分支大量分支时会发生什么？通过在 20 世纪之交询问有关系统过程的问题，数学家开始摸索通往计算世界的道路，但他们还不知道。他们还没有看到延展性和放大的双重荣耀，或与之相关的奥秘。

What happens when the number of steps in an algorithm gets large, the number of loops multiplies, their level of nesting deepens, and the conditional branches ramify vastly? By asking questions about systematic processes at the turn of the twentieth century, mathematicians started to feel their way toward the world of computation, but they didn’t know it yet. They hadn’t yet glimpsed the twin glories of Malleability and Amplification, or the mysteries associated with each.

电子问题

The eProblem

大卫希尔伯特，一位数学家中的王子，在哥廷根大学，数学世界的中心——直到纳粹对犹太数学家的清洗摧毁了它。从 1900 年开始，希尔伯特利用他的国际声望专注于关注具体的难题，其中一些与数学本身的基础有关。这些著名的问题被命名为希尔伯特第二和希尔伯特第十。解决其中任何一个问题都会立即证明您是世界级的数学家。

David Hilbert, a prince among mathematicians, was at the University of Göttingen, a center of the mathematical universe—until the Nazi purge of Jewish mathematicians destroyed it. Beginning in 1900, Hilbert used his international prestige to focus attention on specific hard problems, some of which concerned the very foundations of mathematics itself. These famous problems took names such as Hilbert’s Second and Hilbert’s Tenth. Solving any one of them would immediately establish that you were a world-class mathematician.

1928 年，希尔伯特提出了另一个难题。这是一个简单的逻辑系统——称为一阶逻辑。数学家使用一阶逻辑来制定措辞准确的陈述，这些陈述可能是对的，也可能是错的。希尔伯特问是否有一种系统的方法，一种算法，来决定这个简单逻辑系统中的一个陈述是否正确。

In 1928 Hilbert posed another hard problem. This one was about a simple logic system—called first-order logic. Mathematicians use first-order logic to formulate precisely worded statements, which may be true or false. Hilbert asked if there was a systematic way, an algorithm, to decide if a statement in this simple logic system is true or not.

这是这样一种说法：所有物体都是松果。这是另一个：所有对象都是 quinces。系统中始终允许的复合语句是通过将任意两个其他语句与单词or组合而成的：所有对象都是 pinecones，或者所有对象都是 quinces。系统有一条规则，允许您用重新排列的与其等效的语句替换此组合语句：所有对象都是 pinecones 或 quinces。这里的词等价意思是如果前者为真，则后者为真，如果前者为假，则后者亦为。因此，单词的重新排列不会改变陈述的真值。简单的重新排列将两个最初的陈述折叠成一个陈述，其中或在松果和木瓜之间。这似乎很明显，但关键是这个逻辑系统中的每一步都使用简单甚至微不足道的操作将一个语句转换为等效的语句。新声明源自旧声明。

Here is such a statement: all objects are pinecones. Here’s another: all objects are quinces. A compound statement that is always allowed in the system is made by combining any two other statements with the word or: all objects are pinecones, or all objects are quinces. The system has a rule that lets you replace this combined statement with a rearranged one that’s equivalent to it: all objects are pinecones or quinces. Here the word equivalent means that if the former statement is true, then so is the latter, and if the former is false, then so is the latter. The rearrangement of words, then, doesn’t change the truth value of the statement. The simple rearrangement collapses the two initial statements that were joined by or into one statement with the or between pinecones and quinces instead. This seems obvious, but the point is that each step in this logic system transforms one statement into an equivalent one using a simple, even trivial, operation. The new statement derives from the old one.

推导是此类步骤的序列，其操作就像在每个步骤中应用的or规则一样简单。到目前为止，我们所描述的是一种系统的方法，它可以在每一步都保持真或假的方式来导出一个又一个陈述。如果你从真实的陈述开始，推导总是会产生真实的陈述。

Derivations are sequences of such steps, with operations as simple as the or rule applied at every step. What we’ve described so far is a systematic way to derive one statement after another in such a way that truth, or falsity, is maintained at each step. If you start with true statements, a derivation will always yield a true statement.

这是数学家感兴趣的真实陈述。它们从非常简单的——显然是正确的——陈述开始，并使用逻辑系统从中得出更多真实的陈述。他们开始的真实陈述被称为公理。例如，一个数等于它自己是一个公理。这显然总是正确的。数学令人敬畏的荣耀在于，即使每一步都很简单明了，这种推导也可能导致完全出乎意料的结果。

It’s the true statements that interest mathematicians. They start with very simple—obviously true—statements and derive more true statements from them using the logic system. The true statements they start with are called axioms. For example, it’s an axiom that a number is equal to itself. That’s obviously always true. The awesome glory of math is that such derivations can lead to totally unexpected results, even though every step is simple and obvious.

但希尔伯特要求的是一种决策算法，而不是推导。他并没有要求系统化过程实际上从公理中生成陈述的推导——只是它准确地确定推导是否可能。区别似乎并不重要。如果您可以确定一个陈述是正确的，为什么说明它实际上是如何从公理推导出来的很重要？事实证明，这非常重要。13

But Hilbert asked for a decision algorithm, not a derivation. He did not ask that the systematic process actually generate a derivation of the statement from the axioms—only that it accurately decide whether a derivation was possible or not. The distinction might seem unimportant. If you can decide that a statement is true, why is it important to show how it was actually derived from the axioms? It turns out to be fundamentally important.13

在学术的家族谱系中，我们可以知道，17 世纪的约瑟夫是今天在世的詹姆斯的直系祖先，但他们之间并没有正式建立父子关系，即世代相传。如果它们在 Y（雄性）染色体上共享相同的 DNA——一个简单的实验室测试证实了这一点——那么它们一定是由一个雄性系相关的。它们之间必须存在一条路径。但知道一条路径的存在与了解将特定 DNA 传递给男性线的实际男性系列完全不同——这通常很难确定。知道这样一条路径的存在可能会鼓励您花费研究精力来寻找实际路径——要知道您没有在徒劳的搜索中浪费时间。

In the scholarly genealogy of families, it’s possible to know that Joseph from the seventeenth century, say, was the direct ancestor of James alive today without formally establishing a father-to-son path, a generation-by-generation descent, between them. If they share the same DNA on the Y (male) chromosome—a straightforward lab test establishes this—then they must be related by a male line. A path between them must exist. But knowing that a path exists is nothing at all like knowing the actual series of males who passed the particular DNA down the male line—something that’s often quite difficult to establish. And knowing that such a path exists might encourage you to expend the research effort to find the actual path—to know you weren’t wasting your time in a futile search.

因此，希尔伯特在 1928 年提出的一个大问题是，简单逻辑是否有一个技巧——就像 DNA 测试一样——可以系统地决定一个陈述是否为真，而无需实际从明显为真的公理中推导出来。这被称为希尔伯特的 Entscheidungsproblem。

So Hilbert’s big question of 1928 asked if simple logic had a trick—like the DNA test—that would decide in a systematic way if a statement was true without actually doing the derivation from the obviously true axioms. This is called Hilbert’s Entscheidungsproblem.

这是一个令人生畏的名字，但它只是德语中的决策问题。是否有系统的方法来确定以简单逻辑表达的陈述的真值？然而，把它放在德语中，肯定会提高它对英语读者的深度。决策问题听起来像是一个商学院的话题，但 Entscheidungsproblem 建议一个 Götterdämmerung 来撼动和更新世界，这实际上就是它所做的。让我们称它为eProblem，因为它导致了电子邮件——并且说 Entscheidungsproblem 很快就会变得令人厌烦。

It’s a forbidding name, but it’s simply German for decision problem. Is there a systematic way to decide the truth value of statements expressed in simple logic? Putting it in German, however, surely heightens its profundity for English readers. Decision problem sounds like a business school topic, but Entscheidungsproblem suggests a Götterdämmerung to shake and renew the world, which is actually what it did. Let’s call it the eProblem since it led to email—and saying Entscheidungsproblem soon becomes tiring.

1934 年在英国，Max Newman 在剑桥大学的一次演讲中提出了 eProblem。他说的是一个系统的过程，但“机械过程”是他实际使用的术语。纽曼的措辞是关键。他也可以说“系统过程”、“有效过程”、“配方”或“算法”等等。这个概念还没有确切的词。这正是问题所在。14

In England in 1934, Max Newman presented the eProblem in a lecture at Cambridge University. He talked about a systematic process, but “mechanical process” was the term he actually used. Newman’s choice of words was key. He could just as well have said “systematic process,” “effective process,” “recipe,” or “algorithm,” among others. There were no precise words for the concept yet. That was exactly the problem.14

学生艾伦·图灵参加了那场讲座。极度字面意思的图灵开始用简单的纸“机器”将纽曼的“机械过程”形式化。纽曼肯定对他自己的学生感到惊讶——这么年轻（只有 22 岁）而且笨拙！结结巴巴！——在那场决定性的演讲后不久，用一台简单的机器解决了深刻的电子问题。事实上，纽曼一开始并不相信。这台机器看起来像一个玩具——不是严肃的数学。当然，如此深刻的数学结果不可能来自如此简单的设备。但他很快就对图灵的结果深信不疑。15

Student Alan Turing attended that lecture. The exceedingly literal-minded Turing proceeded to formalize Newman’s “mechanical process” with a simple paper “machine.” Newman was surely surprised that his own student—so young (only 22) and awkward! and stammering!—solved the profound eProblem with a simple machine not long after that fateful lecture. In fact, Newman didn’t believe it at first. The machine seemed like a toy—not serious math. Surely such a profound mathematical result couldn’t have come from such a simple device. But he quickly became convinced of Turing’s result.15

图灵使用他的机器——当然现在称为图灵机——来解决 eProblem。首先，他发明了计算——这正是图灵机所做的。然后他使用计算——对我们所说的系统或机械过程的精确描述——来解决问题。逻辑推导中的简单步骤序列可能会提醒您计算机采取的琐碎步骤的顺序。这两种情况都会从无意识的单个步骤中产生一些注意事项。图灵是第一个正式将两者联系起来的人，这让他成为了数学界的神。

Turing used his machine—now called a Turing machine, of course—to solve the eProblem. First, he invented computation—it’s exactly what Turing machines do. Then he used computation—a precise description of what we mean by systematic, or mechanical, process—to solve the problem. The sequences of simple steps in logical derivations might remind you of the sequences of trivial steps that computers take. Both cases yield something mindful from mindless individual steps. Turing was the first person to formally connect the two, and it made him a god in mathematics.

他表明，简单的逻辑没有技巧 DNA 测试。如果你能找到一个，你必须做完整的推导。不可能有希尔伯特要求的那种算法。正如数学家所说，图灵发现简单的逻辑是不可判定的。如此出乎意料和令人不安的结果，就连伟大的希尔伯特一开始也不相信。如果图灵没有做任何其他事情——比如拯救英国或发明计算——这将使他进入科学万神殿。他解决了其中一个难题。但正是他的机器——而不是他对 eProblem 的解决方案——让他在数学以外的更大世界中出名。现代计算机是图灵机器的直接概念后代。

He showed that there is no trick DNA test for simple logic. You have to do the full derivation, if you can find one. There can’t be an algorithm of the type that Hilbert had asked for. As mathematicians say, Turing found that simple logic is undecidable. It was such an unexpected and disturbing result that even the great Hilbert didn’t believe it at first. If Turing had done nothing else—like save Britain or invent computation—this would have put him in the scientific pantheon. He had solved one of the hard problems. But it was his machine—not his solution of the eProblem—that made him famous in the larger world outside math. The modern computer is a direct conceptual descendant of Turing’s machine.

在图灵解决 eProblem 的同时，新泽西州普林斯顿大学的 Alonzo Church 也在解决这个问题。大约在希尔伯特宣布 eProblem 的时候，美国数学家 Church 已经在哥廷根大学完成了研究生工作。事实上，Church 比 Turing 早了几个月的时间。按照学术界的规则，教会赢了，荣耀通常都是他的。但是图灵的求解技术与丘奇的截然不同。在数学中，证明方法通常与证明本身的事实一样重要。纽曼认为数学界应该知道图灵的新方法。

While Turing was solving the eProblem, so was Alonzo Church at Princeton University in New Jersey. Church, an American mathematician, had done graduate work at the University of Göttingen about the time that Hilbert announced the eProblem. In fact, Church beat Turing to the solution by several months. By the rules of academia Church had won and the glory would have normally been all his. But Turing’s solution technique was strikingly different from Church’s. In math the method of proof is often as important as the fact of the proof itself. Newman thought that the mathematical world should know about Turing’s new method.

纽曼敦促丘奇承认图灵的贡献，丘奇做到了。他们都在 1936 年通过印刷论文上市。这是一大步，因为启发计算机诞生的是图灵的直观、工业、甚至民俗机器，而不是 Church 的深奥形式化（lambda 可定义性）。它们是等价的概念——图灵在他的论文中证明了这一点——但图灵对词语的选择却产生了截然不同的后果。他们俩解决的难题对 Digital Light 来说并不重要，但图灵的解决技术——他的机器——才是。16

Newman urged Church to acknowledge Turing’s contribution, and Church did. They both went public with printed papers in 1936. This was a big step because it was Turing’s intuitive, industrial, even folksy, machine that inspired the birth of the computer, not Church’s abstruse formalization (lambda definability). They were equivalent concepts—Turing proved it in his paper—but Turing’s choice of words had profoundly different consequences. The hard problem that they both solved isn’t important to Digital Light, but Turing’s solution technique—his machine—is.16

这两种方法预示了一场仍然定义计算机科学的小冲突——象牙塔与臭气熏天的战斗。“带着臭味出去”是剑桥大学的俚语，用于获得自然科学学位——化学首当其冲。图灵在剑桥数学的无味、干净的象牙塔中构想了他的计算的伟大想法，但启发计算机的是他坚韧、真实的工业模型，而不是丘奇的纯数学概念。计算机科学仍然沿着这条线分裂。它与一些大学的数学和其他大学的工程混为一谈。延展性是一种塔现象，而放大是一种臭名昭著的荣耀。疯狂冲向第一台计算机的玩家将不得不跨越高塔-臭气熏天的鸿沟。17

The two approaches presaged a skirmish that still defines computer science—the battle of the ivory tower versus the stinks. “Going out in stinks” was slang at Cambridge for taking a degree in the natural sciences—chemistry bearing the brunt of the insult. Turing conceived his great idea of computation in the scentless, clean ivory tower of Cambridge mathematics, but it was his gritty, real, industrial model that inspired the computer, not Church’s pure mathematical concept. Computer science is still split along that line. It’s lumped with mathematics at some universities and engineering at others. Malleability is a tower phenomenon, and Amplification is a stinks glory. The players in the mad rush to the first computers would have to straddle the tower–stinks divide.17

图灵并不是唯一一个将系统过程的概念正式化的人。正如我们所见，Church 也这样做了，还有其他几个。但是图灵的方法是如此有影响力，以至于其他方法相形见绌。在我们的普通世界中，我们仍然使用他在 1936 年的开创性论文中在剑桥引入的概念。他的词可计算是卡住的词。他还给了我们编程的想法，尽管他没有这么称呼它。这使他成为第一个程序员。而且，唉，他是第一个编写有缺陷的软件的人。18

Turing wasn’t the only person to formalize the notion of systematic process. Church did that too, as we’ve seen, and there were several others. But Turing’s method has been so influential that the other approaches pale in comparison. In our ordinary world we still use the concepts that he introduced at Cambridge in his seminal 1936 paper. His word computable is the one that stuck. He also gave us the idea of programming, although he didn’t call it that. That makes him the first programmer. Also, alas, he was the first to write buggy software.18

他是另一个计算机传统的第一个例子。他古怪的性格——强烈的字面意思、对错误的诚实、社交尴尬和对服装的漠视——使他有资格成为第一个极客。也许我们现在会说他“在光谱上”。19

And he was the first example of another computer tradition. His quirky personality—intense literal-mindedness, honesty to a fault, social awkwardness, and sartorial disregard—qualified him as the first geek. Perhaps we would now say he was “on the spectrum.”19

纽曼担心图灵很快就会成为“被证实的孤独者”，并告诉了丘奇。纽曼认为，与普林斯顿大学的逻辑精英——会说同一种语言的人——交流可能会扭转这种趋势。于是他要求丘奇接受图灵为研究生，丘奇再次答应了。图灵将在美国教堂获得博士学位。

Newman was afraid that Turing was fast becoming a “confirmed solitary” and told Church so. Newman thought that mixing with the logical elite at Princeton University—people who could speak the same language—might reverse this tendency. So he asked Church to accept Turing as a graduate student, and Church obliged again. Turing would earn his PhD under Church in America.

图灵乘坐轮船Berengaria前往美国，于 1936 年 9 月在曼哈顿下船。这只是一个巧合，但 Kotelnikov 最近于 5 月降落在那里，也在Berengaria上。俄罗斯人只逗留了 60 天，可能参与了曼哈顿市中心 Amtorg 贸易公司的密码工作。图灵到达时，他已经启程回家了。两艘船在黑暗中经过，可以这么说——在同一艘船上。

Turing sailed to the States on the steamship Berengaria, disembarking in Manhattan in September 1936. It’s just a coincidence, but Kotelnikov had recently landed there, also on the Berengaria, in May. The Russian stayed only 60 days, probably involved in secret code work at the Amtorg Trading Company in downtown Manhattan. He’d already departed for home by the time Turing arrived. Two ships passing in the dark, so to speak—on the same ship.

在这个巧合之后不久就形成了真正的联系。图灵从曼哈顿前往附近的普林斯顿大学学习。然后，他的导师纽曼来到普林斯顿进行了为期六个月的访问，访问了邻近的高等研究院——有时称普林斯学院以将其与大学区分开来（并将演讲者从 IAS 口中拯救出来）。约翰·冯·诺依曼（John von Neumann）——这个故事中的另一位重要人物——已经在王子学院作为其首批永久学者之一。20

A real nexus formed shortly after this coincidence. Turing proceeded from Manhattan to nearby Princeton for his studies there. Then his mentor, Newman, came for a six-month visit to Princeton, to the neighboring Institute for Advanced Study—sometimes called the Princetitute to distinguish it from the University (and to save the speaker from the IAS mouthful). John von Neumann—another major player in this story—was already at the Princetitute as one of its first permanent scholars.20

不是玩具

Not a Toy

要了解图灵的好主意，请考虑图 3.2 中的名片。它有一个角被切除，中间有一个圆孔。正面（有版权）和背面都刻有铭文。您可以欣赏这张卡看似简单的欺骗性，但不要被愚弄——它非常强大。

To get a handle on Turing’s great idea, consider the business card in figure 3.2. It has one corner cut off and a round hole in the center. Both the front (with copyright) and the back are inscribed. You can admire the deceptive simplicity of this card, but don’t be fooled—it’s amazingly powerful.

想象一下卡片后面有一条从左到右的纸带。它被分成方格，通过卡片上的洞你可以看到一个方格。磁带大部分是空白的，但通常有一个或多个带有符号的正方形。在这种情况下，我选择了非空白符号作为数字 1、2、3、4 和 5，但它们也可以是 #、!、$、% 和 &。关键是它们是不同的标记，没有意义。特别是，它们不是数字。我们称它们为符号，但它们没有任何象征意义。只需将 1 替换为 #，将 2 替换为 !，等等，在此名片设备的描述中到处都是，没有任何变化——除了方块上标记的形状。

Imagine that there’s a paper tape running from left to right behind the card. It’s divided into squares, and you can see one square through the hole in the card. The tape is mostly blank, but there are typically one or more squares with symbols on them. I chose the non-blank symbols in this case to be the numerals 1, 2, 3, 4, and 5, but they could just as well be #, !, $, %, and &. The point is that they’re distinct marks, without meaning. In particular, they’re not numbers. We call them symbols, but they symbolize nothing. Simply replace a 1 with #, a 2 with a !, and so on, everywhere in the description of this business card device, and nothing changes—except the shapes of the marks on the squares.

图 3.3，第 1 步（顶行）显示了一个磁带，上面只有四个标记：5155。磁带的其余部分是空白的。该卡显示为略微透明，以便您可以通过它看到胶带。卡片右侧的奇怪标记列表——带有圆圈的数字、冒号、箭头等——包含了管理该设备的六个规则。磁带上每个可能的符号都有一个规则。卡片左侧的倒置列表还有六个规则。

Figure 3.3, step 1 (top row), shows a tape with just four marks on it: 5155. The rest of the tape is blank. The card is shown slightly transparent so that you can see the tape through it. The list of strange markings down the right side of the card—with the circled numbers, colons, arrows, and so on—contains six of the rules that govern this device. There is one rule for each possible symbol on the tape. The upside-down list at the left of the card holds six more rules.

名片设备是这样工作的。找到孔中符号的规则。（不要注意颠倒的规则。）对于第 1 步，它是右上角的规则，即空白处的规则。空格后面有一个冒号，后跟一个 5。它说用 5 替换空格——在空格中写一个 5。5 之后是左箭头。它说将卡片向左移动一格。规则右侧的小字形代表名片本身。它说您应该旋转卡片以匹配字形的方向。在这种情况下，这意味着将卡片顺时针旋转半圈，将裁剪的角放在西北位置，从而产生步骤 2 配置。

The business card device works like this. Find the rule for the symbol in the hole. (Pay no attention to rules written upside down.) For step 1, it’s the rule at top right, the one for the blank. There’s a colon after the blank, followed by a 5. It says to replace the blank with a 5—write a 5 in the blank. Following the 5 is a left arrow. It says to move the card left by one square. The little glyph at the right of the rule represents the business card itself. It says that you should rotate the card to match the glyph’s orientation. In this case, it means to rotate the card one half turn clockwise, putting the cropped corner in the northwest position, resulting in the step 2 configuration.

适用于第 2 步的规则位于右下角，用于孔中的符号 5。它说将 5 更改为 2——擦除 5 然后写一个 2——然后向右移动一格。这次没有小卡片字形，所以卡片保持相同的方向。步骤 3 配置就是结果。

The rule that applies in step 2 is at the lower right, for the symbol 5 in the hole. It says to change the 5 to a 2—erase the 5 then write a 2—then move right one square. There’s no little card glyph this time, so the card remains in the same orientation. The step 3 configuration is the result.

向前跳过两个步骤给出步骤 5 配置（步骤 4 未显示）。在这些步骤中，卡片保持相同的方向（西北）。在这种情况下应用的规则是从右上角开始的第三个规则：将 2 更改为 3，向左移动一格，然后将卡片旋转到所示方向。这里的意思是把卡片翻过来，把裁剪好的角落放在西南。这揭示了卡片的另一面（如步骤 6 所示）和另外两个不同于我们目前看到的两个列表的规则列表。

Skipping ahead two steps gives the step 5 configuration (step 4 not shown). The card maintains the same orientation (northwest) during these steps. The rule to apply in this case is the third one from the upper right: change the 2 to a 3, move left one square, and rotate the card to the orientation shown. Here it means to turn the card over, putting the cropped corner in the southwest. This reveals the other side of the card (shown in step 6) and two more lists of rules different from the two lists we’ve seen so far.

依此类推。如果你进一步追求这种乏味，你会发现这张牌最终会回到原来的方向，洞里有一个 4。适用的规则没有冒号的权利，这意味着没有其他事情发生。计算停止。

And so on and on. If you pursue this tedium further, you’ll find that the card will eventually end up in its original orientation, with a 4 in the hole. The rule that applies has nothing right of the colon, which means nothing else happens. The computation halts.

这不是一场闲置的游戏。不管你信不信，目标是Digital Light。名片设备是图灵机。这是图灵的玩具之一。这是图灵最著名的发明的硬件实现。

This hasn’t been an idle game. The goal, believe it or not, is Digital Light. The business card device is a Turing machine. This is one of Turing’s toys. It’s a hardware implementation of Turing’s most famous invention.

在 1930 年代，“计算机”是一个人——通常是一个女人——执行例如保险公司的簿记过程，或者后来 Bletchley Park 的密码破译过程。在与他的母亲 Sara 的谈话中，图灵将在那里为他工作的数百台“计算机”称为“奴隶”——这表明了系统（或“机械”）过程的惩罚性质。21

In the 1930s a “computer” was a human—usually a woman—who executed the bookkeeping processes of, say, an insurance company or later the codebreaking processes of Bletchley Park. In conversation with his mother Sara, Turing referred to the hundred or so “computers” working for him there as “slaves”—which indicates the punishing nature of systematic (or “mechanical”) processes.21

图灵捕捉到了“计算机”在仔细执行系统过程（例如添加几十个数字的列表）时用铅笔和纸做的事情，也许会用茶歇来打断工作。图灵机是他设计的模型，它捕捉了他们所做工作的本质简单性。

Turing captured what “computers” did with pencil and paper when they carefully executed a systematic process—such as adding a list of, say, several dozen numbers—perhaps interrupting the effort with a tea break. A Turing machine was the model he devised that captured the essential simplicity of what they did.

但它并没有捕捉到他们的乏味。单调乏味——正如你已经发现，如果你遵循了上面的说明——来自于小步骤的不断重复，不断担心错误，以及努力跟踪——比如说，在茶歇期间。它也没有捕捉到他们的无聊。回想一下斯托帕德的汉娜：“无聊呢？瓦尔！这就是你说的吗？” 机器不会感到乏味或无聊。这些都是人的问题。图灵创建了一个抽象模型，说明“计算机”的作用。他把她的乏味从存在中抽象出来，认为是无关紧要的。他坚持诸如不同的心理状态、多个步骤、书写和擦除符号以及无休止的草稿纸供应等概念。

But it didn’t capture their tedium. The tedium comes—as you’ve discovered if you followed the instructions above—from the unceasing repetition of the tiny steps, the constant worry of error, and the effort of keeping track—during a tea break, say. Nor did it capture their boredom. Recall Stoppard’s Hannah: “And the boredom? Val! Is that what you’re saying?” A machine wouldn’t suffer tedium or get bored. Those are human problems. Turing created an abstract model of what a “computer” does that matters. He abstracted her tedium out of existence as immaterial. He maintained notions such as different states of mind, multiple steps, writing and erasing symbols, and an endless supply of scratch paper.

名片可能采用四种可能的方向或状态，并且有六个符号（计算空白）。因此，卡片的每个方向都有六个规则，每个可能出现在孔中的符号一个。一般规则是，每一步孔中的符号可能会改变，孔向左或向右移动一格，卡片方向可能会改变。或者它可能只是停止。这里的所有都是它的。22

There are four possible orientations—or states—the business card may take, and there are six symbols (counting the blank). So, there are six rules for each orientation of the card, one for each symbol that might appear in the hole. The general rule is that at each step the symbol in the hole may change, the hole moves left or right one square, and the card orientation may change. Or it may simply stop. That’s all there is to it.22

图灵将他的每台机器定义为只有四样东西：一维磁带分成正方形，一组有限的符号，一个带有限元的磁带扫描仪状态数，以及一个“指令表”，告诉您如何处理扫描仪状态和磁带符号的每种组合。在我们运行的示例中，磁带扫描仪是一张有孔的名片，六个符号是数字 1 到 5 和一个空白，四个状态是卡片的四个方向。四组规则构成指令表。还有一件事。磁带在任一方向上都尽可能长。如果您需要，总会有另一个广场。总是有更多的草稿纸。也许你现在可以理解为什么纽曼起初无法相信如此简单的设备——它看起来像是一个玩具——会产生一个深刻的数学结果。但是从这个简单的“机器”概念中产生了所有的计算。

Turing defined each of his machines to have only four things: a one-dimensional tape divided into squares, a finite set of symbols for it, a tape scanner with a finite number of states, and an “instruction table” that tells you what to do with each combination of scanner state and tape symbol. In our running example, the tape scanner is the business card with a hole in it, the six symbols are the digits 1 through 5 and a blank, and the four states are the four orientations of the card. The four sets of rules form the instruction table. And one other thing. The tape is as long as necessary in either direction. There’s always another square if you need it. There’s always more scratch paper. Perhaps you can understand now why Newman couldn’t believe at first that such a simple device—a toy it seems—led to a profound mathematical result. But from this simple “machine” concept came all of computing.

您可能会在名片设备中瞥见现代计算机。磁带扫描仪——卡——是 CPU（中央处理器），磁带是内存。但是现代计算机可以执行任何计算，只需更改其程序，无论这意味着什么。当然，我们简单的名片设备不能执行任何计算，不是吗？这是惊喜。是的，它可以。皮克斯可以用它计算玩具总动员！然而，他们不想这样做，因为它太慢了，可能会占用整个宇宙的生命周期，但速度是一个可分离的问题，将在下一章中讨论。关键是我们的设备不仅仅是任何普通的图灵机。名片机是通用的图灵机。23

You might glimpse the modern computer in the business card device. The tape scanner—the card—is the CPU (central processing unit), and the tape is the memory. But a modern computer can execute any computation, by simply changing its program, whatever that means. Surely our simple business card device can’t execute any computation, can it? Here’s the surprise. Yes, it can. Pixar could compute Toy Story with it! They wouldn’t want to, however, because it’s so tediously slow that it might take the lifetime of the universe, but speed is a separable issue, addressed in the next chapter. The point is that our device isn’t just any ordinary Turing machine. The business card machine is a universal Turing machine.23

图灵的伟大想法不仅在于图灵机可以执行系统化过程——我们所说的“系统化过程”或“机械化过程”正是在图灵机中体现出来的。但这本身就是一个很好的主意。加法算法——如何将两个数字相加得到它们的总和——就是这样一个过程。因此，必须有一台图灵机将其磁带上的两个数字相加，并在其磁带上生成它们的总和。并且什么都不做。

Turing’s great idea wasn’t only that a Turing machine could execute a systematic process—that what we mean by a “systematic process” or “mechanical process” is embodied exactly in a Turing machine. But that was a pretty good idea all by itself. The addition algorithm—how to add two numbers together to get their sum—is such a process. So, there has to be a Turing machine that adds two numbers on its tape together and generates their sum on its tape. And does nothing else.

另一种系统过程是反转任何字母串的过程。例如，给定abcdefg，它交换最外面的一对字母，然后是下一个最外面的一对，依此类推。它继续进行，直到没有更多的对可以交换，或者直到只剩下一个字母，在这种情况下产生gfedcba 。因此，存在一台图灵机，它将对其磁带上的任何字符串进行这种反转，而不是其他任何东西。

Another systematic process is one that reverses any string of letters. For example, given abcdefg, it swaps the outermost pair of letters, then the next most outermost pair, and so on. It proceeds until there are no more pairs to swap, or until only a single letter remains, yielding gfedcba in this case. Therefore, a Turing machine exists that will do this reversal for any string on its tape, and nothing else.

然而，图灵的大师之作是展示了一台图灵机如何能够完成任何其他图灵机所能做的事情。它可以执行所有系统过程——包括两个数字的相加或一串字母的反转。它是一台可以计算任何可计算事物的机器。这就是通用在计算中的含义。当我们说图灵的伟大想法是计算时，我们的意思是它是通用计算。现代计算机是这种特殊的图灵机，通用图灵机的后代。但是图灵机怎么能通用呢？

Turing’s master stroke, however, was to show how a single Turing machine can do what any other Turing machine can do. It can perform all systematic processes—including addition of two numbers or reversal of a string of letters. It’s one machine that can compute anything that’s computable. That’s what universal means in computation. When we say Turing’s great idea was computation, we mean it was universal computation. The modern computer is a descendant of this particular kind of Turing machine, the universal Turing machine. But how can a Turing machine be universal?

不要向外行解释计算机。向处女解释性行为更简单。

——罗伯特 A. 海因莱因，《月亮是个严厉的情妇》24

Don’t explain computers to laymen. Simpler to explain sex to a virgin.

—Robert A. Heinlein, The Moon Is a Harsh Mistress24

海因莱因该死！图灵使他的一台机器通用的技巧很聪明。图灵机是由其规则集定义的简单事物。例如，名片机由其 24 条规则定义，卡片的每个方向有 6 条规则。所以图灵推断，必须有可能设计一个图灵机，它可以描述任何图灵机，加上该机器的输入，并模拟该机器在给定磁带的情况下会做什么。他认为这是可能的原因是这样的模拟是一个系统过程，而图灵机旨在捕捉我们所说的系统过程的意思。模拟任何其他机器的图灵机是通用图灵机，图灵展示了如何构建这样的机器。

Heinlein be damned! Turing’s trick for making one of his machines universal is clever. A Turing machine is a simple thing that’s defined by its set of rules. The business card machine, for example, is defined by its 24 rules, 6 for each orientation of the card. So Turing reasoned that it must be possible to design a Turing machine that could take a description of any Turing machine, plus the input to that machine, and simulate what that machine would do given that tape. The reason he believed it to be possible is that such a simulation is a systematic process, and a Turing machine is meant to capture just what we mean by systematic process. A Turing machine that simulates any other is a universal Turing machine, and Turing showed how you could construct such a machine.

图 3.4 中的 A 代表任何图灵机，被描绘为沿着无限长的磁带左右移动的扫描头。名片机就是一个例子。它的“扫描头”就是卡片上的孔。名片将是我们运行任意图灵机 A 的示例。U 是一个通用图灵机，可以模拟任何图灵机 A，给定 A 的规则的一维描述 - 它的指令集 - 和 A 的磁带数据的描述.

The A in figure 3.4 represents any Turing machine, depicted as a scanning head that moves left and right along an infinite tape. The business card machine is an example. Its “scanning head” is the hole in the card. The business card will be our running example of an arbitrary Turing machine A. U is a universal Turing machine that can simulate any Turing machine A, given a one-dimensional description of A’s rules—its instruction set—and a description of A’s tape data.

在我们的示例中，A 的规则就是指定名片机的 24 条规则。它们被编码成UA 要求的格式。名片上的规则是以两张二维指令表的形式写在名片的两侧，其中一半倒置。U 的磁带上需要这些规则的一维版本，写在 U 在那里使用的符号集中。我们将很快展示 A 规则的示例编码。

In our example, A’s rules are just the 24 rules that specify the business card machine. They are encoded into a form required by U. A’s rules as presented on the business card are in the form of two two-dimensional tables of instructions written on two sides of the card, with half of them upside down. U needs a one-dimensional version of these rules on its tape, written in the set of symbols that U uses there. We will show an example encoding of A’s rules shortly.

A 的数据是指最初出现在其磁带上的非空白符号——例如，5155 是前面示例中名片机的初始数据。它也被编码成 U 所需的形式。我们还将展示 A 的数据编码示例。换句话说，通用图灵机磁带上的初始数据 U 由 A 的规则和A 的数据组成，呈一维形式，如图所示。

By A’s data we mean the nonblank symbols that appear initially on its tape—for instance, 5155 was the initial data for the business card machine in the earlier example. It is also encoded into a form required by U. We will show an example of A’s data encoding too. In words, the initial data on the tape of the universal Turing machine, U, consists of both A’s rules and A’s data, in one-dimensional form, as shown.

图灵注意到任何图灵机的规则集都可以写成一行符号。对于名片示例，如果您对名片的四个方向（正面、背面、旋转正面和旋转）使用一个符号缩写 f、b、F 和 B，则可以在一行中列出所有 24 条规则返回——将规则分成这四组，每组有六条规则：

Turing noticed that the set of rules of any Turing machine can be written into one line of symbols. For the business card example, you can list all 24 of its rules in a single line if you use one-symbol abbreviations f, b, F, and B for the four orientations of the card—front, back, rotated front, and rotated back—and divide the rules into those four groups, with six rules per group:

请参阅注释以了解此编码的实际外观。

See the annotation for how this encoding might actually look.

图灵的第一个技巧是将单行机器描述写到 U 的磁带上，每个磁带方格一个符号。把它想象成写在磁带的左半边。

Turing’s first trick was to write that one-line machine description onto U’s tape, one symbol per tape square. Think of it as written in the left half of the tape.

图灵的第二个技巧是在 A 的描述右侧写一个 A 的磁带的单行描述，包含它的初始数据。这里的编码很简单。所以我们的名片机的描述和初始数据可能是这样的，使用竖线分隔两者，使用 0 编码空白：

Turing’s second trick was to write a one-line description of A’s tape, containing its initial data, to the right of A’s description. The encoding here is straightforward. So our business card machine’s description and initial data might look like this, using a vertical bar to separate the two and a 0 to encode a blank:

在这一点上，通用图灵机 U“知道”任意图灵机 A 是什么，因为它具有对该机器行为的完整描述，如其规则所表达的那样。通用机器知道任意机器最初查看的是什么磁带。U 知道它的输入数据。

At this point the universal Turing machine U “knows” what the arbitrary Turing machine A is because it has a complete description of that machine’s behavior, as expressed by its rules. And the universal machine knows what tape that arbitrary machine is initially looking at. U knows its input data.

通用机器只需要知道另外两件事来模拟任意机器 A：它当前正在扫描哪个符号以及它当前处于什么状态。图灵的第三个技巧是在磁带的左侧添加一个符号，指示当前状态和另一个符号右半部分表示当前扫描的方块。关于任意机器 A 的这四条信息——它的描述、初始数据、初始状态和初始扫描符号——形成了通用机器的初始数据。下面是 U 的名片机输入磁带在背面方向的表示，在初始数据 5155 上，最初扫描右侧的空白 (0)：

The universal machine needs to know only two more things to simulate arbitrary machine A: which symbol it’s currently scanning and what state it’s currently in. Turing’s third trick was to add a symbol on the left part of the tape indicating the current state and another symbol on the right half indicating the currently scanned square. These four pieces of information about arbitrary machine A—its description, initial data, initial state, and initially scanned symbol—form the initial data for the universal machine. Here’s a representation of U’s input tape for the business card machine in back orientation, on initial data 5155, and initially scanning the blank (0) just to the right:

粗体0标记最初扫描的符号，粗体b标记卡片的初始方向，因此使用哪些规则。请参阅注释以了解 U 磁带的实际外观。

The bold 0 marks the initially scanned symbol, and the bold b marks the initial orientation of the card, hence which rules to use. See the annotation for how U’s tape might actually look.

为 U 设计一组规则以便它可以模拟任意机器 A 像这样编码到 U 的磁带上是一个乏味的过程。关键是 U 可以“看到”任意机器的完整描述以及该任意机器的数据的完整描述。它可以查看任意机器的当前状态以及该机器的数据磁带当前扫描的方块。这是对要模拟的机器的完整描述。下一刻可以类似地模拟，因此图灵在他 1936 年的著名论文中设计了一个 U，可以系统地将一个时刻的表示转换为下一个时刻的表示。所以后来在模拟名片机的时候，U的磁带是这样的：

The design of a set of rules for U so that it can simulate an arbitrary machine A encoded like this onto U’s tape is a tedious process. The point is that U can “see” a complete description of the arbitrary machine and a complete description of the data for that arbitrary machine. It can see the current state of the arbitrary machine and the currently scanned square of that machine’s data tape. That’s a complete description—at one moment—of the machine to be simulated. The next moment can be similarly simulated, so Turing designed a U in his famous 1936 paper that could systematically convert the representation for one moment into the representation for the next moment. So at some later time in the simulation of the business card machine, U’s tape looks like this:

然后它会模拟下一步，然后是下一步，以此类推。实际的构造是痛苦的，但它的要点并不难。如果您愿意失去海因莱因童贞，请阅读注释以了解模拟的更多详细信息。

Then it would simulate the next step, and then the next step, and so on. The actual construction is painful, but the gist of it isn’t hard. If you’re willing to lose your Heinleinian virginity, read the annotation for further details of the simulation.

图灵展示了机器 U 如何模拟任意机器 A 的每个操作步骤。仅模拟 A 的一个步骤需要很多 U 的步骤，但这对论证无关紧要——速度对于计算的概念并不重要。U 是一台通用计算机，因为它可以计算任何其他机器可以计算的任何东西。只需更改 U 磁带的描述部分即可更改计算的内容。

Turing showed how machine U can simulate each step of operation of an arbitrary machine A. It takes lots of U’s steps to simulate only one of A’s steps, but that’s immaterial to the argument—speed doesn’t matter for the concept of computation. U is a universal computer because it can compute anything that any other machine can compute. Just change the description part of U’s tape to change what gets computed.

图灵发明了编程。用现代术语来说，图灵将任意机器的程序存储在通用机器的内存中，他也将数据存储在那里——通用机器磁带的左右两半。要更改通用机器模拟的机器——即它执行的计算——你只需要更改程序，即磁带左半部分的描述部分。

Turing had invented programming. In modern terminology, Turing stored the program of an arbitrary machine in the memory of the universal machine, and he stored the data for it there too—in the left and right halves of the universal machine’s tape. To change which machine the universal machine simulates—that is, which computation it performs—you need only change the program, the description part in the left half of the tape.

通用图灵机本质上就是我们现在所说的存储程序计算机，因为它以相同的方式存储程序和数据——都在机器的内存中。存储程序计算机就是我们今天所说的单字计算机。我们称任何特定的机器 A，如其编码规则所代表的，计算机程序，或只是程序。这也解释了为什么程序员经常称自己为编码员。他们将任意算法（由特定的图灵机实现）编码为计算机 U 所需的一维形式。图 3.5 显示了通用图灵机 U 如何隐喻地对应于现代计算机。

The universal Turing machine is, in essence, what we now call a stored-program computer, since it stores the program and the data in the same way—both in the memory of the machine. And a stored-program computer is what we mean by the single word computer today. We call any particular machine A, as represented by its encoded rules, a computer program, or just program. This also explains why programmers often call themselves coders. They encode an arbitrary algorithm—implemented by a particular Turing machine—into the one-dimensional form the computer U needs. Figure 3.5 shows how a universal Turing machine U corresponds metaphorically to a modern computer.

现代计算机几乎总是将其程序分成至少两个部分。一部分称为操作系统，或OS，例如 Windows、MacOS 或 AndroidOS。该程序部分始终在运行。这是需要无限循环的情况。为满足您的个人需求而进行更改的程序部分称为应用程序 （或app，流行）。类似地，数据存储器将对操作系统重要的数据与应用程序的数据分开保存。操作系统只是“处理事务”，例如将应用程序加载到内存中的正确位置并启动，处理鼠标或手指输入，以及监视电源故障。这是一个代表现代计算机中任意图灵机 A 的应用程序。每个算法或系统过程都有这样一个 A。这就是计算世界的基础。

A modern computer nearly always has its program divided into at least two parts. One part is called the operating system, or OS, such as Windows, MacOS, or AndroidOS. This program part is always running. It’s a case where an infinite loop is desirable. The program part that gets changed to meet your individual needs is called an application (or app, popularly). Similarly, the data memory holds data of importance to the operating system separate from the data for the app. The operating system just “takes care of business,” like getting an app loaded into memory in the right place and started, taking care of mouse or finger input, and watching for power failure. It’s an app that represents an arbitrary Turing machine A in a modern computer. And there’s such an A for every algorithm, or systematic process. That’s the observation that underlies the world of computation.

值得再次说明图灵的成就。他表明，有一台机器可以做任何系统化的事情。它不能敲钉子或敲击钢琴键，但它可以做任何可以用符号表示的系统操作（然后符号输出可用于驱动敲钉子或敲击琴键的机器）。要改变机器的功能，你所要做的就是改变它的程序或应用程序。他发明了计算机的概念——我们指的是存储程序计算机。我们在硬件中实现它只是为了让它快速运行。

It’s worth stating again what Turing accomplished. He showed that there’s a single machine that can do anything that’s systematic. It can’t hammer nails or strike piano keys, but it can do anything systematic that can be represented with symbols (and then the symbolic output can be used to drive a machine that hammers nails or strikes keys). To change what the machine does, all you have to do is change its program, or app. He invented the concept of computer—by which we mean stored-program computer. We realize it in hardware only to make it go fast.

一台计算机可以计算多少个程序？它可以运行多少个应用程序？嗯，有这么多，你永远无法数完它们。这就像问一架钢琴可以演奏多少首乐曲。计算机是人类发明的最具延展性的工具。这就是延展性的奇妙之处。Digital Light 只是其无限折叠所拥抱的世界之一。

How many programs can a computer compute? How many apps can it run? Well, there are so many that you can’t ever finish counting them. It’s like asking how many pieces of music a piano can play. The computer is the most malleable tool ever invented by humankind. That’s the wonder of Malleability. Digital Light is just one of the worlds embraced in its infinite folds.

约翰尼·冯·诺依曼

Johnny von Neumann

Princetitute——普林斯顿高等研究院的俚语——吸引了天才，尤其是那些逃离纳粹欧洲的天才。它已经收集了一个图灵和纽曼在 1930 年代后期抵达镇上时，这群星团虽小但令人印象深刻，图灵在读研究生，纽曼在休假。最著名的是，阿尔伯特·爱因斯坦已经在那里了。但对于我们的故事来说更重要的是，约翰·冯·诺依曼也是如此。

The Princetitute—slang for the Institute for Advanced Studies in Princeton—was a magnet for geniuses, particularly those fleeing Nazi Europe. It had already collected a small but impressive cluster of stars when Turing and Newman arrived in town in the late 1930s, Turing for his graduate studies and Newman for a sabbatical. Most famously, Albert Einstein was already there. But more importantly for our story, so was John von Neumann.

他于 1903 年 12 月 28 日在布达佩斯出生于 Neumann János Lajos。他的父亲是银行家马克斯·诺伊曼。1913 年，奥匈帝国政府授予马克斯高贵——大概是因为他的经济援助——这使他能够使用冯姓形式。所以马克斯的儿子在美国一直被称为约翰·冯·诺依曼。他于 1933 年永久搬到那里，并成为王子学院最早的成员之一。

He was born Neumann János Lajos on December 28, 1903, in Budapest. His father was Max Neumann, a banker. In 1913 the Austro-Hungarian government ennobled Max—presumably for his financial assistance—which allowed him to use the von surname form. So Max’s son was always known as John von Neumann in America. He moved there permanently in 1933 and became one of the earliest members of the Princetitute.

约翰曾是布达佩斯富裕精英的一员，并在他的新国家继续享受贵族的生活方式。例如，他总是开凯迪拉克，但它们经常被破坏和更换，这是他个性的线索。他回答“约翰尼”，另一个线索。这张约翰尼的著名照片（图 3.6，左上角）显示他身着西装，打着领带——他平时的着装——骑着一头背对着其他人的骡子（你可能会说屁股向后），在开往 Grand 的火车上峡谷。他是一位热情而亲切的主人。他喜欢马提尼酒、可笑的派对帽和一首好色的打油诗。25

John had been a member of the wealthy elite of Budapest and continued to enjoy an aristocratic lifestyle in his new country. He always drove Cadillacs, for example, but they were often wrecked and replaced, a clue to his personality. And he answered to “Johnny,” another clue. This famous photograph of Johnny (figure 3.6, top left) shows him in a business suit and tie—his usual dress—astride a mule that faces away from all the others (ass backward, you might say) in a train headed into the Grand Canyon. He was a bon vivant and gracious host. He loved martinis, ridiculous party hats, and a good salacious limerick.25

冯诺依曼因其对量子物理学和博弈论等多个领域的贡献而闻名，更不用说他在秘密的曼哈顿计划中的成员资格，以发展核武器，并随后成为原子能委员会的成员。但他也因对计算机的贡献而闻名，最著名的是一种被广泛模仿的计算机子系统的通用组织，称为冯诺依曼架构。

Von Neumann became famous for his contributions to fields as diverse as quantum physics and game theory, not to mention his membership in the secretive Manhattan Project for the development of nuclear weapons, and subsequent membership in the Atomic Energy Commission. But he also became famous for his contributions to computers, most notably a widely emulated general organization of computer subsystems called the von Neumann architecture.

他是天才中的天才，是“头脑最快”的“最聪明的人”。他无疑是美国人在计算机竞赛中的常驻天才，他们的对手是英国天才艾伦·图灵。从两者的角度来看，可以很方便地总结一下，Digital Light 的知识遗产来源于存储程序的计算概念，主要归功于 Turing，以及实现该概念的架构，主要归功于 von Neumann。26

He was a genius’s genius, the “cleverest man” with the “fastest mind.” He was certainly the resident genius of the Americans in the race to the computer, their counterpart to the British genius, Alan Turing. To put the two in perspective, it’s convenient to summarize that the intellectual heritage of Digital Light derives from the stored-program conception of computation, principally due to Turing, and from the architecture implementing that conception, principally due to von Neumann.26

冯诺依曼短暂地参与了大卫希尔伯特对数学基础的著名挑战之一。在图灵解决希尔伯特的 Entscheidungsproblem，eProblem 之前几年，他尝试了希尔伯特的第二个问题。

Von Neumann was fleetingly involved with one of David Hilbert’s famous challenges to the foundations of mathematics. He made a stab at Hilbert’s Second Problem several years before Turing tackled Hilbert’s Entscheidungsproblem, the eProblem.

希尔伯特的第二个问题是算术公理——算术的基本，甚至是显而易见的真理——是否一致。重要的是，希尔伯特认为，证明简单的算术——至少——是由一个不会导致矛盾的逻辑系统支持的。这似乎是合理的，但 1931 年库尔特·哥德尔在维也纳证明了这一点一个足够强大以支持算术并且没有矛盾的系统是不完整的。这个令人惊讶的结果，被称为哥德尔不完备定理，意味着在这样一个逻辑系统中必须存在无法在该系统中证明的算术真理。这是另一种说法：一致性的代价要求你满足于不能系统地证明一切。哥德尔的结果表明，不可能有一个通用的数学机器来推导每一个数学真理——这与图灵后来的结果（1936 年）相反，即有一个通用的计算机来处理所有可计算的事情。

Hilbert’s Second asked if the axioms of arithmetic—the fundamental, even obvious, truths of arithmetic—are consistent. It was important, Hilbert thought, to show that simple arithmetic—at the very least—is supported by a logic system that could not lead to contradictions. That seems reasonable, but in 1931 Kurt Gödel in Vienna proved that a system robust enough to support arithmetic and be free of contradictions couldn’t be complete. This astonishing result, known as Gödel’s Incompleteness Theorem, means that there have to be truths about arithmetic in such a logic system that can’t be proved true in that system. Here’s another way to say it: the price of consistency demands that you be content with not being able to prove everything systematically. Gödel’s result said there couldn’t be a universal math machine for deriving every mathematical truth—in contrast to Turing’s later result (1936) that there is a universal computer for everything computable.

哥德尔的结果与数字光的故事密切相关，因为它确立了冯诺依曼的凭据。冯诺依曼实际上是在 1930 年听哥德尔介绍他的革命性成果，在哥德尔发表之前。作为冯诺依曼能力的衡量标准，他几乎立刻就明白了，并直接向哥德尔本人提出了一个更强的结果。我们可以假设，得知哥德尔已经证明了更强的结果，他非常失望。冯·诺依曼性格的另一个一瞥是，从那一刻起，他就停止了对形式逻辑的贡献。如果他不能成为某个领域的佼佼者，那么似乎——如果他不能超越 Gödel Gödel——那么他就会继续统治另一个领域。

Gödel’s result is tangentially germane to the story of Digital Light because it establishes von Neumann’s credentials. Von Neumann actually heard Gödel present his revolutionary result in 1930, before Gödel published it. As a measure of von Neumann’s ability, he almost immediately understood it and proposed a stronger result directly to Gödel himself. He was highly disappointed, we may assume, to learn that Gödel had already proved the stronger result. Another glimpse of von Neumann’s personality is that he ceased contributing to formal logic from that moment on. If he couldn’t be the top dog in a field, it seems—if he couldn’t out-Gödel Gödel—then he moved on to dominate another.

冯诺依曼并没有像图灵那样处理计算，尽管当图灵理论出现时他已经做好充分准备去理解它。相反，他从工程的角度出发，通过考虑实际机器的实现。在开发热核弹或氢弹的过程中，他让自己熟悉了战后 1940 年代美国制造计算机的所有尝试——计算机的直接前身。他正在寻找能够快速进行热核计算的最佳机器。特别是，他在费城找到了一台房间大小的机器，名为 Eniac，建于 1946 年。它类似于我们后来所说的计算机。事实上，他使用 Eniac 进行氢弹计算。这台“几乎是计算机”的缺陷激发了冯诺依曼和他的同事构想出一种计算机的架构——一个带有存储程序的计算机——这将对 Digital Light 的计算机产生重大影响。他们是在恶臭中从必要性中得出的，而图灵在象牙塔中是从理论中得出的。

Von Neumann didn’t approach computation the way that Turing did, although he was fully prepared to understand Turing’s theory when it appeared. Instead he came at it from the engineering angle, by thinking about the realization of actual machines. In his role developing the thermonuclear or H-bomb, he made himself familiar with all the attempts at building calculating machines—immediate predecessors to computing machines—in the United States in the postwar 1940s. He was looking for the best machine to make thermonuclear calculations rapidly. In particular, he found his way to a room-sized machine in Philadelphia, called Eniac, built in 1946. It resembled what we would come to call a computer. Indeed, he used Eniac for H-bomb calculations. The flaws in this “almost-computer” inspired von Neumann and his colleagues to conceive the architecture of a computer—one with a stored program—that would heavily influence the computers of Digital Light. They came at it from necessity in the stinks while Turing did so from theory in the ivory tower.

和图灵一样，冯诺依曼也过着辉煌而短暂的一生。1957 年，53 岁的他被癌症带走，仅仅比图灵晚了三年，因此他也错过了新千年的数字大融合。然而，与图灵不同的是，他的政府在他的有生之年公开庆祝了他。艾森豪威尔总统于 1956 年向他颁发了总统自由勋章。他曾研究过原子弹和氢弹以及美国的洲际弹道导弹战略——他是一个鹰派，没有人怀疑他的忠诚。根据他自己的参议院委员会证词，他是“强烈的反共分子”。但具有讽刺意味的是，他的合作者克劳斯·富克斯将两人开发的秘密信息——一种点燃热核爆炸的方法——传递给了苏联。27

Like Turing, von Neumann led a brilliant but brief life. Cancer took him at age 53 in 1957, just three years after Turing, so he also missed the Great Digital Convergence of the new millennium. Unlike Turing, however, he was openly celebrated by his government in his lifetime. President Eisenhower presented him with the Presidential Medal of Freedom in 1956. He had worked on both the atomic and hydrogen bombs and the ICBM strategy of the United States—and was such a hawk that nobody ever doubted his loyalty. He was “violently anti-communist” by his own Senate committee testimony. But ironically his collaborator Klaus Fuchs passed the secret information that the two developed—a method for igniting thermonuclear explosions—to the Soviets.27

1937 年的某个时刻，冯·诺依曼、马克斯·纽曼和图灵都在普林斯顿——英美计算机事业的未来领袖。1938 年，冯·诺依曼几乎没有错过任何一个节拍，他试图将图灵招募到王子学院。令人惊讶的是，图灵拒绝了这个丰厚的提议。想想看。可能是冯·诺依曼和图灵走出了王子学院的起跑线。第一台计算机可能是美国的。28

There was a moment in 1937 when von Neumann, Max Newman, and Turing were all in Princeton—the future leaders of the British and American computer efforts. Hardly missing a beat, von Neumann attempted to recruit Turing to the Princetitute in 1938. Astonishingly, Turing rejected this plum offer. Think of it. It could have been von Neumann and Turing out of the starting gates at the Princetitute. And the first computer might have been American.28

然而，图灵和冯诺依曼的性格截然相反，以他们两人为主角的团队，极客和快乐生活，可能行不通。冯诺依曼一定已经感觉到他会成为图灵的头号狗。可能图灵也是。然而，这并不能说明问题，因为图灵忠于自己的性格，他自己出击了。他回到英格兰，几乎立即被招募到布莱切利公园。那是 1939 年，英格兰被吓坏了。

Turing’s and von Neumann’s personalities were so diametrically opposed, however, that a team starring the two of them, the geek and the bon vivant, might not have worked. Von Neumann must have sensed he’d be alpha dog to Turing. Probably Turing did too. There’s no telling however because Turing, true to character, struck out on his own. He returned to England, where he was almost immediately recruited into Bletchley Park. It was 1939, and England was frightened for its life.

布莱切利公园

Bletchley Park

布莱切利公园是图灵著名的帮助破解德国人用于战争通信的加密方案的地方。他们使用了一种极其复杂的加密机器，名为 Enigma（它的实际商品名），类似于用木头包裹的老式黑色打字机。职员会在 Enigma 中输入一条短信——比如说从德国海军总部的一位海军上将到海上的一艘 U 型潜艇——机器会以一种方式对这些字母进行加密，然后以第二种方式对被加密的字母进行加密，依此类推。乱码有好几层，每一层都可以改变。U-boat 会像发送者一样配置其接收 Enigma。然后，收到的消息将按相反的顺序逐层解扰，直到原始消息被显示出来并为 U 艇船长输入。他们每天使用一次性垫系统重新配置机器的加扰。回想一下，Kotelnikov 和 Shannon 都证明了这样一个系统是牢不可破的——至少如果使用得当的话。但德国人认为该系统是牢不可破的，因为有很多配置是可能的。事实上，他们从来没有想到图灵和他在布莱切利公园的同事已经打破了它。Bletchley 利用了这样一个事实，即操作员并不总是正确使用该系统，但他们大多使用大量计算。这就是计算的用武之地。（Enigma 在 Bletchley 的小屋 8 中被破解，如图 3.7 中的照片 B 和 C 所示。照片 A 是奇怪的庄园。）但德国人认为该系统是牢不可破的，因为有很多配置是可能的。事实上，他们从来没有想到图灵和他在布莱切利公园的同事已经打破了它。Bletchley 利用了这样一个事实，即操作员并不总是正确使用该系统，但他们大多使用大量计算。这就是计算的用武之地。（Enigma 在 Bletchley 的小屋 8 中被破解，如图 3.7 中的照片 B 和 C 所示。照片 A 是奇怪的庄园。）但德国人认为该系统是牢不可破的，因为有很多配置是可能的。事实上，他们从来没有想到图灵和他在布莱切利公园的同事已经打破了它。Bletchley 利用了这样一个事实，即操作员并不总是正确使用该系统，但他们大多使用大量计算。这就是计算的用武之地。（Enigma 在 Bletchley 的小屋 8 中被破解，如图 3.7 中的照片 B 和 C 所示。照片 A 是奇怪的庄园。）

Bletchley Park was where Turing famously helped crack the encryption scheme that the Germans used for war communications. They employed a devilishly complex encryption machine called Enigma—its actual trade name—which resembled an old-fashioned black typewriter encased in wood. A clerk would type a text message into Enigma—say from an admiral at German naval headquarters to a U-boat at sea—and the machine would scramble the letters one way, then scramble the scrambled letters a second way, and so forth. The scramblings were several layers deep, and each could be changed. The U-boat would configure its receiving Enigma just like the sending one. The received message would then be descrambled layer by layer, in reverse order, until the original message was revealed and typed out for the U-boat skipper. They used a one-time pad system to reconfigure the machine’s scramblings each day. Recall that both Kotelnikov and Shannon had proved such a system was unbreakable—at least if used correctly. But the Germans assumed the system was unbreakable because so many configurations were possible. Indeed, they never figured out that Turing and his colleagues at Bletchley Park had broken it. Bletchley exploited the fact that operators didn’t always use the system properly, but mostly they used massive calculations. That’s where computation comes in. (Enigma was cracked in Bletchley’s Hut 8, shown at photos B and C in figure 3.7. Photo A is the odd manor house.)

解扰 Enigma 代码的试错法是一项非常乏味的工作，最初是由人类手动完成的——由美国数百台计算机的手和大脑完成。人类女性变种，图灵的“奴隶”。为了帮助减轻单调乏味并提高解码速度，布莱切利公园的人建造了大型机器，称为 Bombes。它们还不是计算机，正如我们现在所说的那样，但它们肯定正在走向它们。它们不是可编程的，即使使用硬件切换和电缆，在该术语的任何一般含义中也是如此。从某种意义上说，Bombe 是一台图灵机，但不是通用的图灵机。它执行了一项包含许多步骤的系统任务——尝试了一个 Enigma 可能使用过的无数可能的加扰——但这是它唯一能做的计算。硬件炸弹的速度大大超过了有血有肉的计算机。例如，速度是至关重要的，它可以在 U 艇的路径上向美国和英国商船发送信息。所以推动了——进行计算快。

The trial and error approach to descrambling Enigma code was very tedious work, done at first by hand—by the hands and minds of hundreds of computers of the human female variety, Turing’s “slaves.” To help ease the tedium and to increase the speed of decoding, the Bletchley Park people built large machines, called Bombes. They weren’t computers yet, as we mean that word now, but they were certainly on the way to them. They weren’t programmable, even with hardware toggles and cables, in any general meaning of that term. In a sense a Bombe was a Turing machine, but not a universal Turing machine. It performed a systematic task of many steps—trying out the myriad possible scramblings an Enigma might have used—but that was the only computation it could do. The speed of the hardware Bombes greatly exceeded that of the flesh-and-blood computers. Speed was of the essence, to get messages to American and British merchant ships in the path of a U-boat, say. So the push was on—to make computation fast.

马克斯·纽曼

Max Newman

图灵需要一个合作伙伴，而不是领导者。他太孤独了。这就是马克斯·纽曼（Max Newman）——已经是他的导师和推动者——会再次想到的地方。和图灵一样，纽曼在普林斯顿逗留后回到了英国。与图灵不同的是，他有一个家庭——他的妻子林恩和他们年幼的儿子爱德华和威廉——并且需要保护他们。由于他是犹太人（顺便说一句，冯·诺依曼也是），他担心如果纳粹占领英格兰，他的家人会受到威胁。到 1940 年，他已将林恩和男孩们藏在遥远但熟悉的普林斯顿大学，他们将在那里呆了几年。与此同时，他加入了布莱切利公园。

Turing needed a partner, not a leader. He was too much the loner. That’s where Max Newman—already his mentor and promoter—would figure again. Like Turing, Newman returned to England after his stay in Princeton. Unlike Turing, he had a family—his wife, Lyn, and their young sons, Edward and William—and needed to protect them. Since he was Jewish (as was von Neumann, by the way), he feared for the safety of his family if the Nazis took England. By 1940 he had stashed Lyn and the boys in faraway but familiar Princeton, where they would remain for several years. Meanwhile, he joined Bletchley Park.

图灵用炸弹领导了对 Enigma 加密机的第一波攻击。然后轮到纽曼了。他领导了第二波攻击，针对的是一种较新的德国加密机器——绰号 Tunny（英国的金枪鱼）。这次攻击使用了一台巨大的电子机器 Colossus，它最初是在 1943 年建造的。实际上，使用了十个这样的野兽。所有这些类似于计算机的计算机——它们不是存储程序计算机——在美国几乎计算机的 Eniac 之前都可以运行。29

Turing had led the first-wave attack against the Enigma encryption machine with the Bombes. Then it was Newman’s turn. He led the second-wave attack, against a newer German encryption machine—nicknamed Tunny (British for tuna). This attack employed a giant electronic machine, Colossus, first built in 1943. Actually, ten of these beasts were used. All of these almost-computers—they weren’t stored-program computers—were functional before the almost-computer Eniac in America.29

图灵与 Tunny 及其 Colossi 有很大的间接关系。他提出了一种在布莱切利口中被称为图灵主义的数学见解，这是破解 Tunny 的关键。令人惊讶的是，考虑到他与纽曼的关系以及他自己与 Bombes 的布莱切利公园机器经验，他没有更直接的角色。但他和纽曼当时并没有合作——而且在一段时间内也不会合作——因为解码文本不再让图灵兴奋。他的新兴趣是对声音进行编码。英国政府将他送回美国执行一项特殊任务，从而激发了这种兴趣。30

And Turing indirectly had a lot to do with Tunny and its Colossi. He came up with a mathematical insight known in Bletchley-speak as Turingismus which was key to cracking Tunny. It’s surprising that he didn’t have a more direct part, considering both his relationship with Newman and his own Bletchley Park machine experience with the Bombes. But he and Newman didn’t team up then—and wouldn’t for a while longer—because decoding texts no longer excited Turing. His new interest was encoding voices. The British government ignited that interest by sending him back to America on a special mission.30

声码器连接

The Vocoder Connection

图灵和科捷尔尼科夫于 1936 年在曼哈顿擦肩而过，从未见过面，但他们之间仍然存在令人惊讶的联系。它是声码器——“语音编码器”的缩写——而不是你想象的计算、采样或破译代码。声码器不仅将图灵与科捷利尼科夫纠缠在一起，还与香农甚至索尔仁尼琴纠缠在一起。他们都在声码器上工作。回想起来，与计算机相比，声码器似乎是微不足道的干扰。但在当时，它似乎是一项同样重要的技术。回想起来，我们也可以看到它为未来的光和声数字化进步埋下了种子。

Turing and Kotelnikov had slipped past each other in Manhattan in 1936 and never met, but a surprising connection nevertheless existed between them. It was the vocoder—a contraction of “voice coder”—not computing, sampling, or codebreaking as you might imagine. The vocoder entangled Turing not only with Kotelnikov but also with Shannon and even Solzhenitsyn. They all worked on vocoders. In retrospect the vocoder seems like a trivial distraction compared to the computer. But at the time it appeared to be an equally central piece of technology. In retrospect too, we can see that it held the seeds for future digital advances in light and sound.

回想一下第 2 章，Kotelnikov 和随后的 Shannon 证明一次性垫是牢不可破的。使用一次性填充码增强的声码器使口语与书面语一样安全。一个加密的声码器——一个语音加扰器——就像一个用于语音的 Enigma 机器。

Recall from chapter 2 that Kotelnikov, and subsequently Shannon, proved one-time pads to be unbreakable. A vocoder augmented with one-time-pad codes made the spoken word as secure as the written word. An encrypted vocoder—a voice scrambler—was like an Enigma machine for the voice.

斯大林变得越来越偏执，并要求使用语音扰频器来保护他的通信免受潜在间谍的侵害。这种需求将促使科捷利尼科夫和索尔仁尼琴一起在莫斯科北部的 Marfino sharashka。

Stalin became increasingly paranoid and demanded a voice scrambler to keep his communications safe from potential spies. That demand would drive Kotelnikov and Solzhenitsyn together in the Marfino sharashka in northern Moscow.

丘吉尔和罗斯福（以及后来的杜鲁门）虽然不那么不理性，但也需要一个用于战时通信的语音扰频器——X 系统。这促使图灵和香农于 1943 年在贝尔实验室一起工作。英国政府任命图灵为其专家，以验证 X 系统的安全性。这就是为什么他没有在布莱切利公园与纽曼合作。相反，他与美国政府承担相同任务的专家香农短暂会合。出于安全原因，两人不能提及密码学，但他们可以谈论关于计算、计算机国际象棋和计算机作为人类智能模型的一切。31

Churchill and Roosevelt (and later, Truman), though less irrationally, also needed a voice scrambler for wartime communications—the X System. And that drove Turing and Shannon together at Bell Labs in 1943. The British government appointed Turing as its expert to validate the security of the X System. That’s why he wasn’t teaming up with Newman at Bletchley Park. Instead he joined up briefly with Shannon, the expert with the same assignment from the American government. For security reasons the two couldn’t mention cryptography, but they could talk all they wanted about computation, computer chess, and the computer as a model for human intelligence.31

如此多的像素英雄设计了声码器这一事实是了不起的，但该设备与我们的计算故事真正相关的是图灵自己制作了一个。在他的密友罗宾甘迪的建议下，他称它为“大利拉”——意为“男人的骗子”，暗指参孙的圣经情人和背叛者。设计大利拉让他获得了他所缺乏的动手工程经验。Delilah 不是计算机，但经验让图灵设计出了一台真正的通用图灵机。这是从塔楼到臭气熏天的准备工作。32

The fact that so many of the pixel’s heroes designed vocoders is remarkable, but the real relevance of the device to our computation story is that Turing made one of his own. He called it Delilah—for “man deceiver,” an allusion to the biblical lover and betrayer of Samson—at the suggestion of his close friend, Robin Gandy. Designing Delilah gave him the hands-on engineering experience that he had lacked. Delilah wasn’t a computer, but the experience prepared Turing to design one—a real universal Turing machine. It was a preparatory move from the tower to the stinks.32

有充分的理由进一步研究声码器。这是前两章中频率和采样思想的一个很好的例子。

There’s a good reason to delve further into the vocoder. It’s a good example of the frequency and sampling ideas from the previous two chapters.

首先我们使用傅里叶的频率思想。我们将声音使用的频带分成十个更小的频带。假设人声的频率假设为每秒 0 到 3,000 个周期。（语音通信通常不使用人类听觉能力的全部范围，它可以达到每秒 20,000 个周期。）将此频带视为分为十个带宽频带，每个频带每秒 300 个周期。例如，原始语音消息中小于每秒 300 个周期的所有频率都在频段 1 中，每秒 300 到 600 个周期之间的所有频率都在频段 2 中，依此类推。语音加扰器的粗略想法是以一种已知但秘密的方式对这十个频段进行加扰，传输结果，然后在接收端进行解扰。

First we use Fourier’s frequency idea. We break the band of frequencies used by a voice into say ten smaller bands. Suppose the frequencies assumed for the human voice are 0 to 3,000 cycles per second. (Voice communication typically doesn’t use the full range of human hearing capability, which can reach as high as 20,000 cycles per second.) Think of this band of frequencies as divided into ten bands of bandwidth 300 cycles per second each. For example, all frequencies in the original voice message less than 300 cycles per second are in band 1, those between 300 and 600 cycles per second are in band 2, and so forth. The rough idea of the voice scrambler is to scramble these ten bands in a known but secret way, transmit the result, and then unscramble at the receiving end.

现在我们调用采样定理以适当的采样率用十组样本来表示十个波段中的每一个。实际上被加扰然后传输的是这十组样本。如果一次性填充系统用于额外的安全性，那么它的贡献会在加扰步骤之前添加到每个通道中的样本中。在接收端，频段的解扰发生，并且如果使用一次性填充代码，则减去。根据采样定理重构十组频率分量，然后将它们全部加在一起以获得原始语音消息。

Now we invoke the Sampling Theorem to represent each of the ten bands with ten sets of samples at the appropriate sampling rate. What actually gets scrambled, and then transmitted, are these ten sets of samples. If a one-time pad system is used for additional security, then its contribution is added to the samples in each channel before the scrambling step. At the receiving end, the unscrambling of the bands happens, and the one-time pad code is subtracted if used. The ten sets of frequency components are reconstructed per the Sampling Theorem, and these are then all added back together to get the original voice message.

令人惊讶的是，采样定理被完全使用了。回想一下，那是 1943 年，香农直到 1948 年才出版他的版本。显然，在香农出版五年前，西方完全理解了抽样。采样在俄罗斯被用于他们的声码器也就不足为奇了，因为 Kotelnikov 于 1933 年在那里发表了它。此外，图灵在不久之后在他自己的声码器 Delilah 的设计中使用了采样。他报告说，他在 1943 年第二次从美国回国的船上构思了他的声码器版本。

The surprise in all this is the fact that the Sampling Theorem was being used at all. Recall that this was 1943, and Shannon didn’t publish his version until 1948. Clearly sampling was completely understood in the West five years before Shannon published. It’s no surprise that sampling was used in Russia for their vocoders, because Kotelnikov had published it there in 1933. Furthermore, Turing used sampling in the design of Delilah, his own vocoder, shortly thereafter. He reported that he conceived his version of the vocoder on board the ship he took in 1943 returning home from America the second time.

声码器仍在我们身边，但现在内置了一台计算机。在现代主流音乐中，增强声码器的商标名为 Auto-Tune。今天它是关于语音增强，而不是加密。使用它的音乐家——例如雪儿、劳里·安德森和 T-Pain——是图灵、科捷利尼科夫、香农和索尔仁尼琴的遥远概念表亲和奇怪的伙伴。Auto-Tune 是“人声的 Photoshop”，因为它使许多歌手——不如刚才提到的三位歌手——变得完美。Photoshop 参考提醒我们，像素和索素是同一个想法——我们可以用计算机和光从头开始创建声音模式。

The vocoder is still with us—but now has a computer built into it. In modern mainstream music, the augmented vocoder is trade-named Auto-Tune. Today it’s about voice enhancement, not encryption. Musicians—such as Cher, Laurie Anderson, and T-Pain—who use it are distant conceptual cousins and strange bedfellows of Turing, Kotelnikov, Shannon, and Solzhenitsyn. Auto-Tune is “the Photoshop of the human voice” because it enables many singers—not so good as the three just mentioned—to be pitch perfect. And the Photoshop reference reminds us that pixels and soxels are the same idea—that we can create sound patterns from scratch with computers, as well as light.

不可知性

Unknowability

大多数人认为计算机无法理解，但实际上它们非常简单。您已经体验过名片设备，并使用它进行计算。它只有四个状态和六个符号，而且是纸做的。然而，它可以计算任何可计算的东西。它是一台计算机。

Most people think computers are beyond understanding, but in fact they’re remarkably simple. You’ve already experienced one, the business card device, and computed with it. It has only four states and six symbols, and it’s made of paper. Yet it can compute anything that’s computable. It is a computer.

但是计算机确实必须进行编程。正如您可能怀疑的那样，这是棘手的部分，它可能非常乏味且容易出错。甚至图灵本人也在他的程序中犯了错误。但那是软件，不是硬件。硬件在概念上很简单，并且可以与软件分离。

But computers do have to be programmed. As you might suspect, that’s the tricky part, and it can be quite tedious and error prone. Even Turing himself made mistakes in his programs. But that’s software, not hardware. Hardware is conceptually simple, and it’s separable from software.

了解硬件并不能告诉我们软件。我们可以知道计算机硬件的完整接线图，而无需了解它在计算什么。例如，您现在完全了解名片“硬件”的工作原理。然而，基于这种理解，你不知道它的任何一个软件程序意味着什么。您可以了解施坦威三角钢琴的工作原理，但如果没有乐谱——它的音乐软件，就无法从中演绎出肖邦练习曲。肯定也是确实，您可以在不知道人脑在想什么的情况下知道人脑的完整接线图。

Knowing the hardware doesn’t tell us about the software. We can know the complete wiring diagram of a computer’s hardware without understanding what it’s computing. For example, you now know how the business card “hardware” works—completely. Yet you’ve no clue, based on that understanding, what any one of its software programs means. You can know everything about how a Steinway grand works, but you’ll not deduce a Chopin étude from it without the score—its musical software. Surely it’s also true that you can know the complete wiring diagram of a human brain without knowing what that brain is thinking.

这就是最初激发图灵的难题——eProblem，Hilbert 的 Entscheidungsproblem——发挥作用的地方。需要明确的是，我们在这里主要关心的不是难题——而是计算机，图灵用来解决它的机器。但重要的是要提醒自己，计算的根源有一些深刻的东西。值得提醒的是，很少有人意识到这一点，因此对计算机是什么有错误的直觉。

This is where the hard problem—the eProblem, Hilbert’s Entscheidungsproblem—that originally motivated Turing comes into play. To be clear, our main concern here isn’t the hard problem—it’s the computer, the machine Turing used to solve it. But it’s important to remind ourselves that there’s something profound at the root of computation. It’s worth the reminder because so few people are aware of it and therefore have the wrong intuition about what a computer is.

许多人认为计算机是一台完全确定的机器。这是完全正确的，因为计算机采取的每一步都是完全定义的。例如，名片机的每一步都是由其24条规则的指令表、当前扫描的符号和卡片的当前方向决定的。但是得出的推论——计算机所做的事情因此是完全预先确定的——是错误的。机器的命运是完全确定的，但你不能总是知道那个命运是什么。如果你不能提前知道，那么确定是什么意思？

Many think that a computer is a completely deterministic machine. That’s exactly true because each step that a computer takes is completely defined. For instance, each step of the business card machine is determined by its instruction table of 24 rules, the currently scanned symbol, and the current orientation of the card. But the inference that’s drawn—that what a computer does is therefore completely predetermined—is false. The fate of the machine is completely determined, but you can’t always know what that fate is. And if you can’t know it in advance, then what does determined mean?

要根据机器定义确定，请再次回忆那些多步骤的过程，包括它们的所有循环和分支以及随之而来的自我参考。它们是一种不同的数学动物。毕竟，计算机很困难这种模糊的感觉是有原因的。但这不是因为硬件。这是关于硬件和软件以及一个运行另一个的方式的东西。

To define determined in terms of a machine, recall those many-stepped processes again, with all their loops and branches and attendant self-reference. They’re a different kind of mathematical animal. That vague sense that computers are difficult has something to it, after all. But it’s not because of the hardware. It’s something about hardware and software and the way one runs the other.

请记住，图灵对 eProblem 的解决方案是没有解决方案：简单一阶逻辑中的陈述是真还是假是不可判定的。您当然可以在某些情况下做出决定，但不是全部。没有算法。计算中有一个类似的结果——某种不可知性或不可解性——称为停止问题：一般来说，你甚至无法知道计算是否会停止这样简单的事情！没有系统的测试来决定它是否最终会停止。给定程序及其输入数据，没有任何算法可以发现程序最终会停止还是永远运行。33

Remember that Turing’s solution to the eProblem was that there is no solution: it’s undecidable whether a statement in simple first-order logic is true or false. You can certainly decide in some cases, but not in all. There’s no algorithm for it. There’s a similar consequence in computation—a certain unknowability, or unsolvability—called the halting problem: in general, you cannot even know something as simple as whether a computation will halt! There’s no systematic test to decide if it will eventually stop or not. There’s no algorithm, given a program and its input data, that can discover whether the program will eventually halt or will run forever.33

换句话说，没有用于停止的 DNA 测试。您必须运行该程序才能看到它的作用。也就是说，你必须找到计算机操作从开始到结束的路径——如果有的话。如果它停止了，那么你知道答案，但如果它没有停止，那么你还不知道。如果你让程序运行的时间长一点呢？你可能正在追逐野鹅。它可能处于无限循环中。一般情况下你无法知道。

In other words, there’s no DNA test for halting. You have to run the program to see what it does. That is, you have to find the path of computer operations from the start to the finish—if there is one. If it halts then you know the answer, but if it doesn’t halt, then you don’t yet know. What if you let the program run a little longer? You might be on a wild goose chase. It might be in an infinite loop. You can’t know in general.

所以一台计算机是完全确定在小而在大不可知的。在正常实践中，这种奇怪的行为并不是真正的问题。程序员通常对他们的程序将要做什么有一个很好的想法——只要它正常工作，就是这样。所以在数字光中，不可知主要是理论上的问题。程序员尽量不写不可知的代码。几十年来，许多编程学科已经发展，以帮助他们避免未知的陷阱。

So a computer is completely determined in the small but unknowable in the large. In normal practice this strange behavior isn’t really a problem. Programmers usually have a pretty good idea of what their program is going to do—once it’s working properly, that is. So in Digital Light the unknowable is mostly a theoretical concern. Programmers try their best not to write unknowable code. Many programming disciplines have evolved over the decades to help them avoid the pitfalls of the unknown.

其中一个陷阱是图灵最感兴趣的计算方面的结果：计算在计算时实际改变自身的能力。例如： (1)从输入数y中减去输入数x。(2) 如果结果是否定的，则将步骤 3 中的步骤编号更改为 4，否则将其更改为 5。 (3) 转到步骤 4。 (4) 写入“减号”并停止。(5) 写“加”并停止。

One of the pitfalls is a result of the computing aspect that most interested Turing: the ability for a computation to actually change itself while computing. For example: (1) Subtract input number x from input number y. (2) If the result is negative then change the step number in step 3 to 4 otherwise change it to 5. (3) Go to step 4. (4) Write “minus” and halt. (5) Write “plus” and halt.

让我们尝试将数字 7 分配给x并将 6 分配给y： (1) 7 减去 6 等于 1。 (2) 结果为正，所以： (3) 转到步骤 5。 (5) 写“加号”并停下来。一个程序不仅可以是一个令人讨厌的分支和循环的缠结，而且它可以是一个移动的目标。大多数现代操作系统通过禁止这种自修改代码来“保护”程序员——就像自锁车门一样。它太容易造成破坏。然而，图灵实际上将这个想法带入了真正的计算机世界。当他有机会为实际硬件设计一个条件分支指令时，他的条件分支指令是通过自修改代码实现的。34

Let’s try this with the number 7 assigned to x and 6 assigned to y: (1) 7 minus 6 equals 1. (2) The result is positive, so: (3) Go to step 5. (5) Write “plus” and halt. Not only can a program be a nasty tangle of branches and loops, but it can be a moving target. Most modern operating systems “protect” the programmer—like self-locking car doors—by prohibiting such self-modifying code. It wreaks havoc too easily. Nevertheless, Turing actually pursued the idea into the world of real computers. His conditional branch instruction, when he got a chance to design one for actual hardware, was realized with self-modifying code.34

当您将计算视为人类大脑或思维的模型时，了解这种不可知性是很有用的——就像图灵、冯诺依曼和香农所做的那样。计算结果可能不是一个好的模型，但不能因为声称它是确定性的，因此是预先确定的而立即将其驳回——也就是说，它在某种程度上过于僵化和受限制，不值得我们使用。你只需要让它运行看看它做了什么。

It’s useful to know about that unknowability when you contemplate computation as a model for the human brain or mind—as Turing, von Neumann, and Shannon all did. Computation might not turn out to be a good model, but it can’t be dismissed out of hand by claiming that it’s deterministic and hence predetermined—that is, it’s somehow too rigid and constrained to be worthy of us. You just have to let it run to see what it does.

编程

Programming

编程是计算的秘密，但科学家们至少花了十年时间才意识到这一点。图灵在他 1936 年的著名论文《论可计算数》中展示了第一个程序。他为引入计算的概念机器（图灵机）编写了它们。所以他发明了编程和存储程序的概念。他给了我们计算这个词，但没有给我们编程这个词。这个词是从哪里来的？图灵使用指令表准备来指代编程。

Programming is the secret to computation, but it took at least a decade for scientists to realize that. Turing exhibited the first programs in his famous 1936 paper, “On Computable Numbers.” He wrote them for the conceptual machines—the Turing machines—that introduced computation. So he invented programming and the stored-program concept. He gave us the word computation but not the word programming. Where did that word come from? Turing used instruction table preparation to refer to programming.

让我们更仔细地看看图灵实际上做了什么。假设他想要一台可以反转输入磁带上任何字符串的机器。他设计了一套规则——指令表——用于实现字母系统反转的特定图灵机。称它为 A。然后他会将 A 的规则交给一台通用计算机。回想一下在其磁带上显示任意机器 A 和在其磁带上显示通用机器 U 的图片。图中的“A 的编码规则”就是我们现在所说的A程序。聪明而有趣的部分是为 A 设计指令表，而不是将其编码为 U 所需的形式。编码很简单，而且，机械的。在我们的名片机示例中，用 0 代替空白，用 f 代替正面方向，等等。很蠢的东西。机器不会觉得无聊，但人类会。

Let’s look more closely at what Turing actually did. Suppose he wanted a machine that reverses any string of letters on its input tape. He designed the set of rules—the instruction table—for a particular Turing machine that implemented a systematic reversal of letters. Call it A. Then he would hand A’s rules to a universal computer. Recall the picture that shows an arbitrary machine A on its tape and the universal machine U on its tape. “A’s encoded rules” in that picture is what we would now call the program for A. The clever, intellectually interesting part is designing the instruction table for A—not encoding it into the form required by U. Encoding is straightforward and, well, mechanical. In our example of the business card machine, substitute a 0 for a blank, an f for front orientation, and so on. Pretty dumb stuff. A machine wouldn’t get bored doing it, but a human would.

将编程分为创意部分和无聊部分仍然存在。在钢琴的隐喻中，乐曲的组成是创造性的部分，而将其编码为音符和乐谱上的休止符是无聊的部分。现代计算机的一个主要用途是为计算执行无聊的编码步骤。它称为汇编或编译。程序员在高符号级别（使用类似英语的语言）工作以创建程序，这是有趣的部分。然后计算机将程序编码成计算机实际理解的冗长乏味的长序列，几乎是微不足道的操作——顺便说一下，这是计算机的常见非数字使用的一个很好的例子。无论如何，这就是它现在的工作方式。

The separation of programming into the creative part and the boring part still holds. In the piano metaphor, the composition of a piece is the creative part, and its encoding as notes and rests on a musical staff is the boring bit. One principal use of modern computers is to do that boring encoding step for computations. It’s called assembly or compiling. A programmer works at a high symbolic level—in an English-like language—to create a program, the fun part. Then a computer encodes the program into those tedious long sequences of almost trivial operations that a computer actually understands—a great example, by the way, of a common nonnumeric use of computers. That’s the way it works now anyway.

然而，在 1940 年代后期，那些准备制造第一台硬件计算机的人使用“设置”一词来描述您如何让机器计算您希望它计算的内容。在几乎是计算机的早期——例如，像 Eniac——这意味着实际插入电缆和切换开关，因为这是可以在硬件上安装程序的唯一方式。后来——正如对存储程序计算机的设想——“设置”似乎不仅意味着创建一个程序，而且确保它被加载到计算机内存中的正确位置。

In the late 1940s, however, those preparing to build the first hardware computers used the term “setting up” to describe how you made a machine compute what you wanted it to compute. In the slightly earlier days of the almost-computers—like Eniac, for example—it meant actually inserting cables and toggling switches, since that was the only way a program could be installed on the hardware. Later—as stored-program computers were contemplated—“setting up” seems to have meant not only creating a program but making sure it was loaded into the memory of the computer in the right place.

起初，它就像是事后的想法——现在我们有了一台机器，让我们开始吧。那时，仅仅制造一台可以工作的计算机就几乎占用了工程师的所有创造性带宽。但他们很快发现“设置”是一项复杂、容易出错且繁重的活动。它变成了摇狗的尾巴。现在这似乎很明显，因为有些程序有数百名程序员编写的数百万个步骤。

At first it was like an afterthought—now that we have a machine, let’s make it go. Back then just building a computer that worked took nearly all the creative bandwidth of engineers. But they soon discovered that “setting up” was a complex, error-prone, and burdensome activity. It became the tail that wagged the dog. This seems obvious now, when some programs have millions of steps written by hundreds of programmers.

但当时，他们需要一套技术来指导和简化“设置”任务，他们也需要一个更好的名称。Princetitute 的冯诺依曼档案中的文件记录了动词to program首次出现的那一刻。它既是编程这个词的诞生，也是编程艺术，一门新学科。36

But back then, they needed a body of techniques to guide and ease the task of “setting up,” and they needed a better name for it, too. Documents in the von Neumann archive at the Princetitute capture the very moment when the verb to program first appeared. It’s both the birth of the word programming and the art of programming, a new discipline.36

在 1945 年 9 月 5 日的一份备忘录中，每个地方编程都是一个有用的术语，冯诺依曼改为使用“设置”，用占位符围绕它引号。他说：“我想再次强调这一点。. . 如此灵活且高度自动化的“设置”机器解决问题的系统对于人们应该考虑的科学用途来说是绝对必要的。” 他说，无论我们怎么称呼它，这都是绝对必要的。37

In a memo dated September 5, 1945, every place programming would have been a useful term, von Neumann instead used “setting up,” surrounding it with placeholding quotation marks. He said, “I wish to reemphasize that . . . such a flexible and highly automated system of ‘setting up’ the machine for a problem [is] absolutely necessary for the scientific uses which one should contemplate.” Whatever we call it, he said, it’s absolutely necessary.37

一封日期为 11 月 1 日的信在保护性引号内暂时建议了“设置”的替代方案：“我们计划的电子精密设备当然会在速度、灵活性（通用字符）和至少会像“设置”或“编程”一样容易。”也就是说，他说，它不能像 Eniac 那样是拨动开关和电缆。38

A letter dated November 1 tentatively suggests, within protective quotation marks, an alternative to “set up”: “The electronic precision device which we are planning would of course surpass it [Eniac] in speed, and flexibility (all-purpose character) and will be at least as easy to ‘set up’ or ‘program.’” That is, he said, it can’t be toggles and cables as it was with Eniac.38

然后它发生了。仅仅几周后的 11 月 19 日，“设置”就再也没有出现过。它现在正在编程，没有带歉意的引号。第三份文件是在项目成员 Vladimir Zworykin 办公室举行的会议记录。“下表列出的代码只是为了证明可以完成这项工作。列出的操作足以完成编程。” 在这几分钟里，不仅编程，而且编码。39

Then it happened. Just a couple of weeks later on November 19, “setting up” never appeared. It was programming now, and without apologetic quotation marks. The third document is the minutes of a meeting held in the office of a project member, Vladimir Zworykin. “The code outlined in the following table is presented simply to prove that it is possible to do the job. The operations listed are sufficient for complete programming.” Not only programming but coding was prefigured in these minutes.39

冯诺依曼团队显然意味着编程是创造性的部分，而编码是无聊的部分，是一项“秘书任务”。今天的程序员称自己为编码员，他们编写代码。它们指的是创造性方面，无论是什么术语。他们明白，无聊的部分现在已归入机器本身。

The von Neumann team apparently meant programming to be the creative part and coding to be the boring part, a “secretarial task.” Programmers today call themselves coders, and they write code. They mean the creative aspect, whatever the term. They understand that the boring part is now relegated to the machine itself.

所以在 1945 年末，冯·诺依曼团队，可能是冯·诺依曼本人，是第一个或多或少地按照我们今天的意思使用编程这个词的人。图灵很快也在 1947 年的一次演讲中使用了这个术语，但他在有限的上下文中只使用了几次。40

So in late 1945 the von Neumann team, probably von Neumann himself, was the first to use the word programming more or less the way we mean it today. Turing soon used the term too, in a lecture in 1947, but he used it only a few times in a constrained context.40

不管是什么术语，很早就清楚地表明，编程，创造性的方面，是计算的困难部分，并且必须发展它的一门学科。世界各地的计算机科学系都从这种认识中脱颖而出。

Whatever the term, it became clear fairly early that programming, the creative aspect, was the difficult part of computing, and that a discipline for it would have to develop. Computer science departments all over the world sprang from that recognition.

编程是制作代表虚构世界的像素的方法。它将 Digital Light 从仅仅拍摄照片扩展到制作它们——从仅仅拍摄它们到计算它们。

Programming is the way to make pixels that represent fictitious worlds. It extends Digital Light from just taking pictures to making them—from just shooting them to computing them.

关于计算机的神话

Myths about Computers

下一章是关于将计算概念转化为实际实践——第一台机器的竞赛。在我们到达那里之前，让我们消除关于计算机的三个神话：它们不一定是电子的，它们不一定是由比特组成的，它们不是基于数字的——甚至不是 0 和 1。

The next chapter is about turning the concept of computation into real practice—the race for the first machine. Before we get there, let’s dispel three myths about computers: they don’t have to be electronic, they aren’t necessarily made of bits, and they aren’t based on numbers—not even 0s and 1s.

首先，计算机不必看起来像您通常知道的任何东西：您的手机、笔记本电脑、台式机、大公司的巨型主机或高科技研究机构的超级计算机。名片设备就是一个很好的反例。它是一台计算机，为了满足我在本书中的目的，它被实现为纸上墨水、硬卡片纸和薄金属。这显然不是电子的。

First of all, computers don’t have to look anything at all like those you commonly know: your cellphone, laptop, desktop, the giant mainframe at your large corporation, or the supercomputer at your high-tech research facility. The business card device is a good counterexample. It’s a computer, and to serve my purposes in this book, it’s been realized as ink on paper, stiff card stock, and thin metal. It’s obviously not electronic.

一个流行的教学技巧是让教室里的学生变成一台计算机。让我们把这个想法提升一个档次。假设我们让美国所有 12 岁以上的人排成一排。那条线将成为我们计算机的磁带——它的内存。如果我们需要更多记忆，我们会招募一些加拿大人和墨西哥人。假设每个人有五顶不同颜色的帽子。这五顶彩色帽子将成为我们磁带上的符号。第六个符号将是没有帽子，这将是除了少数人之外的所有人的默认条件。我们将选择一个人作为磁带的扫描仪。当我第一次设计这个例子时，巴拉克奥巴马是总统，所以我们会让他成为我们的磁带扫描仪，让他从磁带正方形的职责中解脱出来。他随身带着名片机的 24 条规则——它的指令集——用五种颜色而不是数字 1 到 5 表示（空白没有帽子）。规矩照常分为四组规矩，每组规矩都在他的一个口袋里，两个在前面，两个在后面。所以他的口袋是四个状态。

A popular pedagogical trick is to have a classroom of students become a computer. Let’s take the idea up a notch. Suppose we get every person in the United States who’s older than 12 to line up in a row. That line is going to be the tape of our computer—its memory. We’ll enlist some Canadians and Mexicans if we need more memory. Suppose each person has five hats of different colors. Those five colored hats will be the symbols on our tape. A sixth symbol will be the absence of a hat, and that’ll be the default condition of all except a few people. We’ll select one person to be the scanner of the tape. Barack Obama was president when I first devised this example, so we’ll make him our tape scanner, relieving him from tape-square duty. He has the 24 rules of the business card machine with him—its instruction set—expressed in the five colors rather than the numerals 1 through 5 (and no hat for the blank). The rules are divided up into four sets of rules as usual, and each set of rules is in one of his pockets, two in the front and two in the back. So his pockets are the four states.

奥巴马从一个戴红帽子的人开始，然后应用一套装在口袋里的规则——比如说他右前口袋里的规则。如果奥巴马右前口袋里的红帽规则规定，被称呼的人会将帽子换成另一种颜色，或者完全摘下帽子。然后奥巴马根据相同的规则向左或向右移动一个人。然后，如果当前规则指示他，他会为下一条规则更换口袋。等等。那是一台由人类组成的计算机，带有用于状态的口袋和用于符号的帽子。它可以计算任何东西，因为它与名片机、通用计算机做同样的事情。这只是同一台机器的不同硬件实现。但它是一台真正的计算机——当然不是电子的（或数字的）。

Obama starts with a person wearing a red hat and applies one of the pocketed sets of rules—say the rules in his right front pocket. The person addressed will swap hats to one of a different color, or remove his hat entirely, if the rule in Obama’s right front pocket for red hats says to. Then Obama moves left or right one person, according to the same rule. He then changes pockets for the next rule if the current rule instructs him to. And so forth. That’s a computer made of humans, with pockets for states and hats for symbols. It can compute anything, because it’s doing the same thing as the business card machine, a universal computer. It’s just a different hardware realization of the same machine. But it’s a real computer—and certainly not electronic (or numeric).

名片计算机，或者它的人类等价物，也是我们第二个被揭穿的关于计算机的信念的反例——它们必须由比特组成。回想一下第 2 章，一个位有两种状态，就像一个可以向上或向下的电灯开关。但是名片机有四种状态，或者说方向。每个磁带方块可以包含六个值或符号。名片机中没有一个只有两个值的部分。它不是由比特组成的。也不是人类的等价物。

The business card computer, or its human equivalent, is also a counterexample to our second debunked belief about computers—that they must be composed of bits. Recall from chapter 2 that a bit has two states, like a light switch that can be either up or down. But the business card machine has four states, or orientations. And each tape square can hold six values, or symbols. Nowhere in the business card machine is there a part with just two values. It’s not made of bits. Neither is the human equivalent.

位不是必需的，但工程师很快发现它们在实践中非常方便。事实上，他们在计算机设计中使用了多年正式将图灵的伟大想法转化为比特。那个人就是克劳德·香农。也许是他 1943 年在曼哈顿与图灵的会面——他们表面上以声码器为导向的会议——引发了这种兴趣。香农表明，具有任意数量符号的图灵机可以被另一个只有两个符号的图灵机替换，并且将计算相同的东西。由于图灵机的磁带是它的内存，香农的等效图灵机中的磁带方块是一个内存位，两个符号作为值。42

Bits aren’t required, but engineers quickly discovered that they are extremely handy in practice. In fact, they used bits in computer design for years before someone got around to formally converting Turing’s great idea to bits. That someone was Claude Shannon. Perhaps it was his 1943 meetings with Turing in Manhattan—their ostensibly vocoder-oriented meetings—that kicked off this interest. Shannon showed that a Turing machine with any number of symbols can be replaced with another that has only two symbols and will compute the same thing. Since the tape of a Turing machine is its memory, a tape square in Shannon’s equivalent Turing machine is one bit of memory, with the two symbols as values.42

第三个神话更有害。相信位的两个值必须是 0 和 1。换句话说，第三个神话是计算机是由 0 和 1 构成的，因此计算机从根本上讲是关于数字的。这个神话将计算器（一种用于数字运算的机器）与计算机混为一谈，这是一个更大的想法。它还将字形（名称）的概念与其可能表示的内容（即，与命名的内容）混为一谈。

The third myth is more pernicious. It’s the belief that the two values of a bit must be 0 and 1. In other words, the third myth is that computers are built of 0s and 1s and that therefore computers are fundamentally about numbers. This myth conflates a calculator, which is a machine for number crunching, with a computer, a much larger idea. It also conflates the notion of a glyph (a name) with what it might represent (that is, with what is named).

这种误解是很自然的，源于我们通常为比特的两种状态选择名称。我们方便地称它们为 0 和 1。但我们可以称它们为昼夜、向下和向上、点和破折号，或者 ooh 和 aah。我想你会同意这些名称不如 0 和 1 方便或简洁。但是，与名片机一样，它们只是名称而不是数字。

The misconception is a natural one, arising from our usual choice of names for the two states of a bit. We conveniently call them 0 and 1. But we could call them night and day, down and up, dot and dash, or ooh and aah. I think you’ll agree that those names aren’t as convenient, or succinct certainly, as 0 and 1. But, as with the business card machine, they’re just names and not numbers.

很容易理解为什么计算会与数字混淆。图灵的原始论文的标题是“关于可计算数”。Eniac 中的“n”代表“数字”。算法一词向早期的算术致敬。当时在非常慢的机器上进行的许多原始计算都是数字化的——例如早期的氢弹计算。数字是新的计算工具最容易攻击的第一个符号。一些早期的实践者自己似乎从来没有理解计算机不仅仅是一个快速的数字运算器——一个计算器——但图灵知道计算机还能做多少。事实上，有趣的是，布莱切利公园的图灵炸弹和纽曼巨像并没有计算或操纵数字，而是使用符号替换来破解密码。43

It’s easy to understand why computation has been confused with numbers. Turing’s original paper was titled “On Computable Numbers.” The “n” in Eniac stands for “numerical.” The word algorithm pays tribute to early arithmetic. Many of the original computations on the very slow machines of the time were numeric—such as the early H-bomb calculations. Numbers were the easiest first symbols to attack with the new tool of computation. Some of the early practitioners themselves seem never to have understood that the computer was much more than a fast number cruncher—a calculator—but Turing knew how much more computers could do. In fact, it’s interesting to note that Turing’s Bombes and Newman’s Colossi at Bletchley Park didn’t calculate or manipulate numbers but instead used symbol substitution to break codes.43

本节题词的作者查尔斯·谢灵顿爵士使用他著名的“魔法织机”比喻来描述人类大脑的清醒皮层，但它也适合现代计算机的工作原理。这是关于改变模式。这与数字无关，除非我们赋予模式的意义是数字。但这只是我们可以对这些模式做出的无数解释之一。对计算机来说，它只是模式。

Sir Charles Sherrington, the author of this section’s epigraph, used his famous “enchanted loom” metaphor to describe the waking cortex of the human brain, but it also fits the workings of a modern computer. It’s about changing patterns. It’s not about numbers unless the meaning we attach to the patterns is number. But that’s only one of myriad interpretations we can make of the patterns. To the computer it’s just pattern.

如果计算机中没有数字，那还有什么？今天世界上几乎所有的计算机都是电子的并且是由比特构成的。它们是由计算机芯片制成的，这也许是我们最伟大的技术成就。如果你能在这样的芯片内部看一下，你会发现高电压和低电压的模式。那些是位。

If there are no numbers in a computer, then what is there? Nearly all the world’s computers today are electronic and built of bits. They’re built from computer chips, perhaps our greatest technological achievement. If you could look inside such a chip in operation, you’d find patterns of high and low electric voltages. Those are the bits.

一个您可以看到且具有相似模式的熟悉的地方——好吧，无论如何都要测量——是你家中的电源插座。我们知道那里的交流电压以傅里叶波的形式从最大电压波峰变化到最小电压波谷每秒 60 个周期（美国）。通过您的插头进入您的计算机实际上是一个不断变化的波，从低电压到高电压再返回。

A familiar place with a similar pattern that you can see—well, measure anyway—is an electrical outlet in your home. We know that the AC voltage there varies in a Fourier wave from a maximum voltage crest to a minimum voltage trough 60 cycles per second (US). What comes through your plug and into your computer is actually a constantly varying wave, going from low to high voltage and back again.

但是在芯片中，电压不会像在交流电源插座上那样平稳且不间断地变化。它保持其值（例如，高），直到计算机规则指示它更改为另一个值（低）。每个芯片位置都可以采用高电压或低电压——并在控制下切换到另一个——实现了位的抽象概念。我们所说的是，在指示更改为 0 之前，位是 1。看看这些名称有多方便！但物理现实是电压——就像交流电源插座上的现实——而不是数字。

But in a chip, voltage doesn’t vary smoothly and incessantly the way it does at an AC outlet. It maintains its value—say, high—until it’s instructed by a computer rule to change to the other value, low. Each chip location that can assume either a high or low voltage—and change to the other under control—realizes the abstract idea of bit. What we say is that a bit is a 1 until instructed to change to a 0. See how convenient those names are! But the physical reality is voltages—like the reality at an AC outlet—not numbers.

如果您考虑一台内置数万亿位的计算机，通常以常规数组的形式，那么您就会明白为什么 Sherrington 的比喻有效。当它以当今计算机的惊人速度运行时，数以万亿计的电压模式在每秒变化数十亿次的电子舞蹈中扭曲和编织。有时他们用整块布编织电影。

If you think about a computer with trillions of bits built into it, often in regular arrays, then you’ll see why Sherrington’s metaphor works. When it’s operating at the tremendous speeds of today’s computers, the patterns of trillions of voltages warp and weave in an electronic dance that changes billions of times per second. Sometimes they weave movies out of whole cloth.

Digital Light 的秘密不在于您可以通过某种方式将图片转化为数字，就像旧的“数字绘画”涂色书一样。就是我们每天看到的视觉模式，明暗和运动的模式，可以用机器中的电压模式来表示。我们用计算机的模式模仿世界的模式。

The secret of Digital Light isn’t that somehow you can turn pictures into numbers, like the old “paint by numbers” coloring books. It’s that the visual patterns we see every day, patterns of light and shade and movement, can be represented by the voltage patterns in a machine. We mimic the patterns of the world with the patterns of our computers.

把它绑在一起

Tying It All Together

对于 Bletchley Park 的密码学家和数学家小组来说，文本是加密的德国战争信息。对于伦敦著名的布卢姆斯伯里作家和思想家群体来说，这是高级英国文学。乍一看，布莱切利和布卢姆斯伯里似乎没有什么共同之处。但这两个圈子在 1940 年代和 1950 年代以松散但奇怪的方式相互交织在一起。

To the Bletchley Park group of cryptographers and mathematicians, text was encrypted German war messages. To London’s illustrious Bloomsbury Group of writers and thinkers, it was high English literature. It would appear at first blush that Bletchley and Bloomsbury had little in common. But the two circles intertwined with one another in loose but curious ways in the 1940s and 1950s.

这并不意外，因为英国知识分子，无论是数学还是文学，都倾向于通过剑桥或牛津进行筛选。例如，小说家 EM Forster、出版商伦纳德·伍尔夫和经济学家约翰·梅纳德·凯恩斯都是著名的布卢姆斯伯里成员。他们都属于剑桥的秘密社团使徒，其中包括艾伦·图灵的合作者罗宾·甘迪和大卫·尚佩诺恩。

This is not unexpected since British intellectuals, whether mathematical or literary, tend to filter through either Cambridge or Oxford. For example, the novelist E. M. Forster, the publisher Leonard Woolf, and the economist John Maynard Keynes were all renowned Bloomsbury members. And they all belonged to Cambridge’s secret society, the Apostles, which included Alan Turing’s collaborators, Robin Gandy and David Champernowne.

然而，这是巧合。离它不远的是通过 Stracheys 的家庭联系。Bloomsbury 的 Lytton Strachey 有一个兄弟 Oliver，他是 Bletchley Park 的密码学家，还有一个侄子 Christopher，Oliver 的儿子，他是 Turing 在剑桥的同学。克里斯托弗·斯特拉奇（Christopher Strachey）在下一章中作为第一个有记录的电子游戏的作者而引人注目。

That’s coincidence, however. Not far removed from it is the familial connections via the Stracheys. Bloomsbury’s Lytton Strachey had a brother, Oliver, who was a cryptographer at Bletchley Park, and a nephew, Christopher, Oliver’s son, who was Turing’s classmate at Cambridge. Christopher Strachey is notable in the next chapter as author of the first documented videogame.

但真正有意义的联系将布卢姆斯伯里的福斯特与图灵联系在一起。福斯特的小说莫里斯敢于传达——仅在手稿中——成为“奥斯卡王尔德那种难以言喻的人”的感觉。据他的传记作者说，图灵在这本书中得到了安慰。44

But a truly meaningful connection tied Bloomsbury’s Forster to Turing. Forster’s novel Maurice dared to convey—in manuscript only—what it was like to be “an unmentionable of the Oscar Wilde sort.” Turing took comfort in this book, according to his biographer.44

也许最有趣的领带关系到另一位布鲁姆斯伯里巨人弗吉尼亚·伍尔夫，她的丈夫伦纳德也是她的出版商。弗吉尼亚以保存大量日记而闻名，因此伦纳德的故事编辑之一林恩·欧文的名字出现在其中也就不足为奇了。出人意料的转折是林恩嫁给了图灵的导师和布莱切利公园的贡献者非文学作品马克斯纽曼。45

Perhaps the most interesting tie concerns yet another Bloomsbury giant, Virginia Woolf, whose husband, Leonard, was also her publisher. Virginia famously kept extensive diaries so it’s not surprising that the name of one of Leonard’s story editors, Lyn Irvine, appears in them. The unexpected twist is that Lyn married non-literary Max Newman, Turing’s mentor and fellow Bletchley Park contributor.45

在战争期间担心他们的半犹太人儿子，林恩与新泽西州普林斯顿的男孩们一起流亡。马克斯从战前与图灵一起度过的那段时间就知道该地区。但这一次他留在了英格兰，只留下林恩和孩子们在一起。在一次国事访问中，约翰梅纳德凯恩斯在华盛顿与罗斯福总统呆了几个小时后，在普林斯顿对她进行了搜查。他的举动让她耳目一新，让她想起了伦敦，也让她从智力孤立中解脱出来。

Fearing for their half-Jewish sons during the war, Lyn took exile with the boys in Princeton, New Jersey. Max knew the area from the time he spent there with Turing before the war. But he remained in England this time, leaving Lyn alone with the children. On a state visit, after spending several hours with President Roosevelt in Washington, John Maynard Keynes searched her out in Princeton. His gesture was a refreshing reminder to her of London and a respite from intellectual isolation.

林恩和马克斯的小儿子威廉记得这次访问。凯恩斯陪年轻的威廉去理发店理发，让他坐在冯诺依曼旁边。但有些事情让男孩心烦意乱，他哭着逃走了。凯恩斯在冯诺依曼刮胡子的时候，用银色的舌头追着并哄着他回到椅子上理发。46

Lyn and Max’s younger son, William, remembered the visit. Keynes accompanied young William to a barbershop for a haircut and seated him next to von Neumann. But something upset the boy, who fled in tears. Keynes pursued and coaxed him, with silvery tongue no doubt, back into the chair for his haircut while von Neumann got a shave.46

林恩与图灵的友谊后来开始了，当时两人都回到了英国并住在曼彻斯特。仍然渴望伦敦的文学界，她发现他与马克斯的其他主要谈论数学的来访者不同，令人耳目一新。她认为他是一个“非常简单、谦逊、温和的人”。我们期待她对圣图灵有更深入的了解。

Lyn’s friendship with Turing began later when both were back in England and living in Manchester. Still starved for the London literary scene, she found him a refreshing change from Max’s other visitors who talked mostly of math. She saw him as a “very simple, humble, gentle personality.” We look to her for a deeper understanding of St. Turing.

虽然在她眼中算不上极客，但她确实在印刷品上触及了图灵的极客特征。她在萨拉·图灵关于她儿子的传记的前言中写道：“他的衣服从来都不好看。” 并且，“他有一种奇怪的不让人看到眼睛的方式。”

Although hardly a geek in her eyes, she did touch on Turing’s geeky characteristics in print. She wrote in the foreword to Sara Turing’s biography of her son, “He never looked right in his clothes.” And, “He had a strange way of not meeting the eye.”

林恩在被捕后与图灵变得特别亲近，除了官方机密外，他再也没有什么可隐瞒的了。在最后的日子里，他对她说：“我简直不敢相信和一个女孩上床就像和一个男孩上床一样好。”

Lyn became particularly close to Turing after he was arrested and no longer had anything to hide—other than official secrets. He said to her during those last days, “I just can’t believe it’s as nice to go to bed with a girl as with a boy.”

“我完全同意你的看法，”林回答道。“我也更喜欢男孩。”

“I entirely agree with you,” Lyn replied. “I also much prefer boys.”

他们之间产生了一种与怪胎不相称的亲密关系。现在，他的眼睛不再是“一种不符合眼睛的奇怪方式”，而是“蓝色到彩色玻璃的亮度和丰富性，”她写道。“但一旦他以亲切交谈的自信，直接而认真地看着他的同伴，他的眼睛就再也不能错过了。这样的坦率和理解，从他们身上看，是一种文明到让人几乎不敢呼吸的东西。” 像图灵的母亲一样，林恩从不相信他是自杀的。47

There grew between them an intimacy that belied the geek. Now, instead of “a strange way of not meeting the eye,” his eyes were “blue to the brightness and richness of stained glass,” she wrote. “But once he had looked directly and earnestly at his companion, in the confidence of friendly talk, his eyes could never again be missed. Such candour and comprehension looked from them, something so civilized that one hardly dared to breathe.” Like Turing’s mother, Lyn never believed he committed suicide.47

但林恩对 Digital Light 的持久贡献是她的儿子。威廉纽曼将合着第一本计算机图形学教科书。

But Lyn’s lasting contribution to Digital Light was her son. William Newman would coauthor the first textbook on computer graphics.

去比赛

Off to the Races

当我们准备开始第 4 章关于数字光的诞生以及它如何与计算机的诞生交织在一起时，这里是对第 3 章的快速回顾。我通过在小范围内强调其无意识的简单性，在直观的概念层面上介绍了计算并比较其令人敬畏的延展性和理论上的奥秘——它的不可知性——在大范围内。我建议编程是创造力和困难所在。我将计算确立为操纵符号模式的系统或算法过程。这些过程是通过非常简单的单个步骤来实现的——但其中有很多。我打消了我们认为数字是计算的基础的观念——计算机中没有 0 和 1——并且确定性在某种程度上限制了这个想法。我展示了计算不需要速度、比特和电子设备。

As we get set to start chapter 4 about the birth of Digital Light—and how it’s intertwined with the birth of computers—here’s a quick review of chapter 3. I presented computation at an intuitive, conceptual level by stressing its mindless simplicity in the small and comparing its awesome Malleability and theoretical mystery—its unknowability—in the large. I suggested that programming is where the creativity—and difficulty—lies. I established computation as systematic, or algorithmic, processes that manipulate patterns of symbols. The processes are realized with ever-so-simple individual steps—but lots of them. I disabused us of the notion that number is fundamental to computation—there are no 0s and 1s in a computer—and that determinism somehow limits the idea. I showed that speed, bits, and electronics aren’t required for computation.

然而，下一章是关于速度、比特和电子学的。速度在现实世界中非常重要，而位和电子设备是实现它的关键。意想不到的礼物是令人敬畏的放大及其数量级的奥秘，计算进入超新星并发出数字光。

Nevertheless, the next chapter is about speed, bits, and electronics. Speed matters a great deal in the real world, and bits and electronics are key to achieving it. The unexpected gift is awesome Amplification and its order-of-magnitude mystery, where computation goes supernova and emits Digital Light.

黎明前：美国佬对英国人

Predawn: The Yanks versus the Brits

这是一场古老的比赛——至少自美国独立战争以来。刻板印象维持它：美国佬夸大其词，而英国人则轻描淡写。或者更糟糕的是：美国佬使用美元，而英国人使用大脑。英国间谍小说家约翰·勒卡雷 (John le Carré) 曾将美国人称为表兄弟，而他并不是指接吻类。表兄弟是反对希特勒和冷战的盟友，但合作往往不情愿。两支队伍在争夺第一台计算机的过程中既竞争又合作。这是一场势均力敌的比赛——如此之近，以至于一些历史学家仍在争论。

It’s an old competition—since the American Revolution at least. Stereotypes maintain it: Yanks overclaim while Brits understate. Or worse: Yanks use bucks where Brits use brains. The British spy novelist John le Carré famously called Americans the Cousins, and he didn’t mean the kissing variety. The Cousins were allies in the war against Hitler and then the Cold War, but cooperation was often reluctant. The two teams both competed and cooperated in the race to the first computer. And it was a close race—so close that some historians are still debating it.

在 1960 年代，我这一代人被教导说，1945 年出生在费城的 Eniac——一台由 Yanks（我们）制造的机器——是第一台电子计算机。但它不是——不是按照今天计算机作为电子存储程序机器的含义。它甚至不是第一台几乎计算机。英国人（他们）已经赢得了那轮比赛，但我们洋基队却不知道。有那个官方保密法——图灵的暴君——又上场了。英国人在洋基队甚至没有意识到的情况下赢得了第一次小规模冲突。

In the 1960s my American generation was taught that Eniac, born in Philadelphia in 1945—a machine made by Yanks (us)—was the first electronic computer. But it wasn’t—not by today’s meaning of computer as an electronic stored-program machine. It wasn’t even the first of the almost-computers. The Brits (them) had already won that round, but we Yanks couldn’t have known it. There’s that Official Secrets Act—Turing’s tyrant—in play again. The Brits had won the first skirmish without the Yanks even being aware of it.

在发表不知情或不准确的声明时，英国人也不是无可指责的。他们声称 1944 年初在布莱切利公园的机器 Colossus 是第一台电子计算机。那也是错误的。Colossus 和 Eniac 一样，使用的是硬件程序，而不是软件。两者都很大，都是电子的，但它们不是存储程序的计算机。重新编程这两种野兽都需要切换开关和重新插入电缆——以人类的速度手动重新布线机器硬件。图灵存储程序计算机思想的美妙之处在于程序本身可以存储在保存其数据的同一内存中。更改程序及其操作的数据是相同的过程。如果计算机是电子的，那么改变程序的速度就会以电子速度发生，这会让人类望而却步。3

The Brits aren’t blameless either, when it comes to making uninformed or inaccurate pronouncements. They’ve claimed that their machine Colossus at Bletchley Park in early 1944 was the first electronic computer. That’s wrong too. Colossus, like Eniac, used hardware programs, not software. Both were big and both were electronic, but they weren’t stored-program computers. Reprogramming either beast required toggling switches and replugging cables—a manual rewiring of the machine hardware at pokey human speeds. The beauty of Turing’s stored-program computer idea is that a program itself can be stored in the very same memory that holds its data. To change a program and the data it manipulates are the same process. If the computer is electronic, then changing programs happens at electronic speeds that leave humans in the dust. You don’t need an external guy—or, more often in fact, gal—to slowly swap out cables and throw switches.3

但那些房间大小的庞然大物为表兄弟们服务得很好。几乎是计算机在军事上增强了洋基人和英国人。Eniac 进行了氢弹计算。Colossus 帮助破解了德国的加密方案，并确保诺曼底登陆成功。对于将着手设计和制造真正的计算机的工程师来说，这些机器是非凡的电子训练场——或至少是灵感。那些真正的计算机将由存储在计算机中的软件程序驱动，就像数据一样，而不是由电缆从外部强加。4

But those room-sized behemoths served the Cousins well. The almost-computers boosted both the Yanks and the Brits militarily. Eniac made H-bomb calculations. Colossus helped to crack a German encryption scheme and ensured a successful D-Day landing in Normandy. The machines were extraordinary electronic training grounds—or inspirations, at least—for the engineers who would proceed to design and build true computers. Those true computers would be driven by software programs stored in the computer just like data, not imposed from the outside by a cable gal.4

曙光和计算机

The Dawning of the Light—and Computers

弗雷迪·威廉姆斯于 1946 年 12 月搬到曼彻斯特大学。他的第一个目标不是制造一台计算机，而是一台快速的内存。他在战时雷达方面的经验——在屏幕上显示光点——激发了他的想法，即最好的解决方案可能在于阴极射线管。1946 年，他在纽约贝尔实验室看到了一种很有前途的雷达技术。5

Freddie Williams moved to the University of Manchester in December 1946. His first goal wasn’t to build a computer but rather a fast memory for one. His experience with wartime radar—featuring blips on a screen—inspired his idea that the best solution probably lay with the cathode-ray tube. He had seen a promising radar technique at Bell Labs in New York when he visited in 1946.5

就在他搬到曼彻斯特之前，威廉姆斯成功地在 CRT 上存储了一个位。换句话说，他设计了一种方法，可以在管子扁平端的屏幕上的某个位置显示一个小斑点（一个点）或一个大斑点（一个破折号）。他可以写一点。此外，他（嗯，他的电子电路）可以查看已经存储在管表面上的斑点，并确定它是大还是小——他可以读取存储在那里的一点。他也可以将一个大斑点变成一个小斑点，反之亦然。用方便的现代术语来说，他可以将 0 或 1 任意写入一个位置，并且他可以读取那里存储的内容。最重要的是，他写到一个位置的值一直存在，直到他或计算机明确更改它。换句话说，它被记住了。瞧！1946 年，威廉姆斯创造了第一个可见的计算机位。6

Just before his move to Manchester, Williams had succeeded in storing one bit on a CRT. He had devised, in other words, a way to display a small blob (a dot) or a big blob (a dash) at a location on the screen at the flattened end of the tube. He could write a bit. Furthermore, he (well, his electronic circuits) could look at a blob that was already stored on the face of the tube and determine whether it was big or small—he could read a bit stored there. He could also change a large blob to a small one, or vice versa. In convenient modern terminology, he could arbitrarily write a 0 or 1 to a location, and he could read what was stored there. Most importantly, a value he wrote to a location persisted there until he, or a computer, explicitly changed it. In other words, it was remembered. Voila! In 1946 Williams had created the first visible computer bit.6

他正在着手创造所谓的威廉姆斯管。它应该被称为 Williams-Kilburn 管——但几乎从来没有——因为 Tom Kilburn 很快就在曼彻斯特加入了威廉姆斯，并且到 1947 年 3 月已经积极参与管设计。到年底，Kilburn 能够以矩形阵列的形式在管表面存储 1,024 位，并创建并拍摄了 First Light 的照片，这是第一个数字光显示器（图 4.2）。然后他很快就提高到 2,048 位，创建了另一张照片，并拍摄了照片（图 4.3）。但是这个词还不存在，正如我们从第 3 章中知道的那样，所以他使用了digit反而。他写了一篇报告，他的博士论文，“用于二进制数字计算机的存储系统”，发表于 1947 年 12 月 1 日。他 26 岁。那是那个时刻。

He was on his way to creating what would be called a Williams tube. It should be called a Williams-Kilburn tube—but hardly ever is—because Tom Kilburn soon joined Williams at Manchester and by March 1947 was already actively involved in the tube design. By the end of the year Kilburn was able to store 1,024 bits on the face of the tube in a rectangular array—and to create and take the photograph of First Light, that first Digital Light display (figure 4.2). Then quickly he advanced to 2,048 bits, created another picture, and took its photograph (figure 4.3). But the word bit didn’t exist yet, as we know from chapter 3, so he used digit instead. He wrote a report, his doctoral thesis, “A Storage System for Use with Binary Digital Computing Machines,” issued December 1, 1947. He was 26. That was the moment.

基尔本 1947 年的报告用两张照片公开宣布了数码光的曙光。它们不仅仅是任何一系列景点的照片。它们是两张以数字方式渲染并存储在电子数字存储器中的图片。Kilburn打算将它们做成二维图片。

Kilburn’s 1947 report publicly announced the dawn of Digital Light with the two photographs. They weren’t photos of just any array of spots. They were two pictures rendered digitally and stored in an electronic digital memory. Kilburn intended them to be two-dimensional pictures.

Kilburn 通过手工为 1,024 个光点中的每一个进行照明的“过于费力”的方法，创作了第一张数码照片 First Light。显示器是一个由 32 x 32 扩展像素组成的阵列。他使用“以打字机的形式排列”的 32 个键来切换位。Kilburn 将显示器描述为一组“图片元素”。正如我们所见，短格式像素还不存在——直到 1965 年。他将显示元素（可见）与图片元素（不可见）混淆了。我们在 Kotelnikov 章节中讨论了这种区别，但对于 Kilburn 来说还没有这种区别。7

Kilburn created First Light, the first digital picture, by the “far too laborious” method of lighting each of 1,024 spots by hand. The display was an array of 32 by 32 spread pixels. He used 32 keys “arranged in the form of a typewriter” to toggle in the bits. Kilburn described the display as an array of “picture elements.” The short form pixel didn’t yet exist—not until 1965, as we’ve seen. And he confused display elements (visible) with picture elements (invisible). We discussed this distinction in the Kotelnikov chapter, but there was no such distinction yet for Kilburn.7

Kilburn 使用同样繁琐的类似打字机的方法在 2,048 位内存中创建了第二张数字图片。这些图片中的每个散布像素都有两种状态，破折号（大斑点）和点（小斑点），方便地称为 1 和 0。这里的散布像素是圆形飞溅或斑点，具有两种不同大小的光。即使在最早的日子里，也没有小广场。

Kilburn created the second digital picture in the 2,048-bit memory using the same tedious typewriter-like method. Each spread pixel in these pictures had two states, dash (big blob) and dot (little blob), known conveniently as 1 and 0. A spread pixel here is a rounded splash, or blob, of light of two different sizes. There were no little squares—even in the earliest days.

Kilburn 的目的——在他的报告和标题中明确说明——是为计算机设计内存。他在报告中描述了这样一台假设的计算机。然后他和威廉姆斯立即着手建造一台真正的计算机，并在半年内交付了著名的婴儿。8

Kilburn’s purpose—clearly stated in his report and in its title—was to design a memory for a computer. He described just such a hypothetical computer in the report. Then he and Williams promptly proceeded to build an actual computer, and within half a year delivered the famous Baby.8

在图 4.4 中，清醒的、适当奖励的 Kilburn 是皇家学会会员，他拿着他的名声标志，一个 Williams 管——一个 Williams-Kilburn 管，确切地说是笨拙的——而 Baby 则在背景中徘徊。威廉姆斯和基尔本在他们的一生中都获得了巨大的荣誉。到 1977 年弗雷迪·威廉姆斯去世时，他已经是弗雷德里克爵士，也是皇家学会会员。基尔本于 2001 年去世一年后加利福尼亚的计算机历史博物馆让他成为研究员，这是新千年的合适开始。

The sober, suitably rewarded Kilburn in figure 4.4, a Fellow of the Royal Society, holds an icon of his fame, a Williams tube—a Williams-Kilburn tube to be clumsily exact—while Baby hovers in the background. Williams and Kilburn both received great honors during their lifetime. By the time Freddie Williams died in 1977, he was Sir Frederic and also a Fellow of the Royal Society. Kilburn died in 2001 one year after the Computer History Museum in California made him a Fellow, a fitting beginning of the new millennium.

禁果

Forbidden Fruit

但没有证据表明 Kilburn 曾经了解计算机上的第一张照片 First Light 的重要性。据我们所知，他在 1947 年创作了前两张照片后，就再也没有创作过另一张照片。事实上，也没有其他人在 Baby 上拍过照片。人们似乎觉得，拍照并不是对计算机的认真使用。这是亵渎神明的。那是禁果——虽然不完全是雕刻的图像。战争刚刚结束，计算资源稀缺，对它们的需求非常强烈。与 Williams 和 Kilburn 合作开发 Baby 的 David “Dai” Edwards 表示，当时用户对这款机器“绝望”。团队的另一名成员 Geoff Tootill 表示，当时团队仍然在战时精神下运作，没有时间放松。9

But there’s no evidence that Kilburn ever understood the momentousness of First Light, the first picture on a computer. After he created the first two pictures in 1947, he never made another one, so far as we know. In fact, nobody else made pictures on Baby either. People seem to have felt that picture making just wasn’t a serious use of the computer. It was profane. It was forbidden fruit—although not exactly a graven image. The war was just over, computation resources were scarce, and the need for them was intense. David “Dai” Edwards, who worked with Williams and Kilburn on Baby, said that users were “desperate” for the machine then. Another member of the team, Geoff Tootill, said that the team still operated under a wartime ethos then and had no time for relaxing.9

拍照对他们来说是一种轻松、愉快的活动。一张图片作为记忆测试模式是一回事，否则为了这种轻浮和世俗的追求而浪费宝贝稀有的计算资源就完全是另一回事了。相反，机器应该计算一些“严肃”的东西。

Making pictures was a relaxing, good-time activity to them. A picture as a memory test pattern was one thing, but otherwise wasting Baby’s rare computing resource for such a frivolous and profane pursuit was something else altogether. Instead, the machine should be computing something “serious.”

从角度来看，考虑几十年后完全不同的环境，即 1998 年，婴儿的精确复制品是为 50 周年而建造的。庆祝活动的编程竞赛引发了几个 Baby 程序，这些程序在 Baby 的显示器上创建了图片，甚至是动画图片。一生成词BABY。

For perspective, consider the totally different environment decades later, in 1998, when an exact replica of Baby was built for its fiftieth anniversary. A programming contest for the celebration elicited several Baby programs that created pictures, even animated ones, on Baby’s monitor. One generated the word BABY.

婴儿重建由克里斯·伯顿在曼彻斯特带头。2013 年 7 月 4 日，他安排我去曼彻斯特科学与工业博物馆看 Baby。BabyPIXAR滚动显示器（图 4.5）——一个电脑动画来欢迎我！我的主持人 Brian Mulholland 前一天为它编写了 Baby 程序。但它本可以在 1948 年在 Baby 身上运行。10

The Baby rebuild had been spearheaded in Manchester by Chris Burton. He arranged a visit for me to see Baby at Manchester’s Museum of Science & Industry on July 4, 2013. Baby welcomed me with PIXAR scrolling across its monitor (figure 4.5)—a computer animation! My host, Brian Mulholland, had written the Baby program for it the previous day. But it could have run on Baby in 1948.10

对于那些安于和平时期、充斥着计算资源并且更熟悉计算机作为图片制作者而不是数字处理器的现代程序员来说，图片创作是显而易见的——甚至是不可抗拒的。没有视觉显示、动画和交互的计算机现在几乎闻所未闻。我们称它们为服务器。

Picture creation is obvious—irresistible even—for modern programmers who are comfortable in peacetime, flush with computational resources, and more familiar with computers as picture producers than as number crunchers. Computers without a visual display, animation, and interaction are almost unheard of now. We call them servers instead.

比赛

The Race

梳理计算机的早期历史仍然不容易。即使是曼彻斯特婴儿的单一故事也很复杂。完整的历史将有太多人、机器和国家——以及太多的细微差别——不能舒适地放在一本书中，更不用说一章了。我仔细定义了我们所说的计算机的含义，并只关注那些导致数字光的途径，从而修剪掉了大部分的灌木丛。这使得早期计算机历史的一个非常强大的部分完好无损。

It’s still not easy to sort out the early history of computers. Even the single story of Baby at Manchester is complicated. A complete history would have too many people, machines, and countries—and too much nuance—to fit comfortably in a book, much less a chapter. I prune away much of this thicket with a careful definition of what we mean by a computer and by focusing only on those pathways that led to Digital Light. This leaves intact a remarkably robust part of early computer history.

然而，那个较小的部分仍然有数十名玩家。为了提供结构，我继续将他们分成两支球队——英国队和洋基队——并假设他们之间有一场戏剧性的比赛。这是一部奥林匹克小说。参赛者不再是国家队，也不是奥运代表队——只受他们挥舞的旗帜和奖牌数量的约束。然而，该设备组织和跟踪复杂的奥运会，它也适用于计算机历史。竞争背后也有一些真相。

Yet that smaller part still has dozens of players. To give structure I continue to divide them into two teams—the Brits and the Yanks—and posit a dramatic race between them. It’s an Olympian fiction. The competitors were no more national teams than Olympic teams are—bound only by the flag they wave and the medal count. Nevertheless, the device organizes and tracks the complex Olympic games, and it works for computer history too. And there’s a bit of truth behind the competition too.

计算机早期历史的“流程图”（图 4.6）给予硬件和软件贡献者同等的赞誉。计算机几乎从未出现过这种情况——这反映了永无止境的高塔与臭气熏天的战斗。这里的软件被概括为包括所有仅是精神上的工作。

A “flow chart” (figure 4.6) of the early history of computers gives hardware and software contributors equal credit. This has hardly ever been the case for computers—a reflection of the never-ending tower versus stinks battle. Software here is generalized to include all work that is only mental.

同时也是著名理论家的软件贡献者——特别是图灵和冯诺依曼——经常被授予荣誉，但事实并非如此。Baby 的硬件“母亲”Williams 和 Kilburn（实线）对 Baby 本身来说比其早期的程序员 Turing 和 Newman（虚线）更重要。

Software contributors who were also famous theoreticians—Turing and von Neumann, notably—often got assigned the glory when it just didn’t happen that way. Baby’s hardware “mothers,” Williams and Kilburn (solid lines), were more important to Baby itself than its early programmers, Turing and Newman (dashed lines).

流程图也让各个参与者之间的相互联系更加明显。人用圆圈表示，计算机用矩形表示，概念用平行四边形表示。图表中没有可行的直线叙述路径。但它是有根的。最重要和最中心的是图灵 1936 年的“论可计算数”，该概念文件以计算的定义和对存储程序计算机的详细描述开始了这一切。从中流出了冯诺依曼 1945 年的Edvac 报告和图灵 1946 年的Ace 报告，其中描述了实际计算机的两种不同架构。

The flow chart also makes the interconnectedness of the various players obvious. People are represented by circles, computers by rectangles, and concepts by parallelograms. There’s no viable straight-line narrative path through the chart. But it is rooted. Topmost and center is Turing’s 1936 “On Computable Numbers,” the concept paper that started it all with its definition of computation and its careful description of a stored-program computer. From it flowed von Neumann’s 1945 Edvac report and Turing’s 1946 Ace report, which described two different architectures for actual computers.

编程是最著名的软件形式。计算的数学是另一回事，如“论可计算数”。第三种脑力劳动以 Edvac 和 Ace 报告为代表。因此，完全意义上的软件贡献包括存储程序计算机的概念、一个架构或它的程序。图表中 Edvac 报告的虚线显示了它对图表中几乎所有计算机的概念（软）贡献，除了图灵的直线。

Programming is the best-known form of software. The mathematics of computation is another, as in “On Computable Numbers.” The third type of mental work is represented by the Edvac and Ace reports. A software contribution in its full sense therefore includes the idea of the stored-program computer, an architecture for one, or programs for it. The dotted lines flowing from the Edvac report in the chart show its conceptual (soft) contribution to almost every computer in the chart aside from Turing’s direct line.

同样，但在硬件方面，Williams 和 Kilburn 影响了两个团队的几台早期机器的设计。它们的设计（硬）贡献由流程图中每个的实线表示。他们不是任何其他非曼彻斯特计算机的工程师，但 Williams-Kilburn 管直接影响了这些计算机的设计。

Similarly, but in hardware, Williams and Kilburn influenced the design of several early machines on both teams. Their design (hard) contribution is indicated by solid lines from each in the flow chart. They weren’t engineers on any of the other non-Manchester computers, but the Williams-Kilburn tube directly influenced the design of those machines.

建筑与设计

Architecture versus Design

听起来，计算机架构是一个计划。它与计算机设计明显不同。我将保留这个熟悉的词，以使架构存在于现实世界中，使用实际的电子设备。建筑是理念；设计就是现实。设计是一个硬件问题，至少需要与建筑一样多的创造力——有些人会说得更多。但这是一种不同的创造力。那是因为架构并不规定或指导设计。一所房子的建筑可以体现在许多设计中。一个标有厨房的盒子激发了——但绝不是指导——花岗岩、瓷砖、木材、把手、灯、水龙头、油漆、窗扇等的无数组合。

A computer architecture is—as it sounds—a plan. It differs markedly from a computer design. I’ll reserve that otherwise familiar word for the act of making an architecture exist in the real world, with actual electronic devices. Architecture is idea; design is reality. Design is a hardware issue, requiring at least as much creativity as architecture—some would say more. But it’s a different kind of creativity. That’s because architecture does not dictate or instruct design. One house architecture can be manifested in many designs. A box labeled Kitchen inspires—but in no way directs—a myriad combinations of granites, tiles, woods, handles, lights, faucets, paints, casements, and so forth.

Edvac 报告中指定的著名架构——称为冯诺依曼架构——至少影响了流程图中显示的六种不同的计算机设计，甚至更多。图 4.7 显示了它的典型平面图。考虑标记为Memory的简单框。没有指定实际的内存技术——这是硬件设计问题。

The famous architecture specified in the Edvac report—called the von Neumann architecture—influenced at least the six different computer designs shown in the flow chart, and many more. Figure 4.7 shows it in typical plan view. Consider the simple box labeled Memory. There’s no actual memory technology specified—that’s a hardware design concern.

设计人员必须拿出那个空盒子，并弄清楚如何在现实世界中制造一个能够正确地将那个盒子连接到计划中的其他空盒子的设备。然后他们必须掌握他们选择的物理特性，并使所有设备与一个真正的内存一起工作，该内存保存实际的比特，与自身外部的世界通信，并在给定的电网上工作。从建筑中的空盒子到设计的步骤是一个重要的创造性步骤。它没有算法。

The designers have to take that empty box and figure out how to make a device in the real world that will correctly connect that box to the other empty boxes in the plan. Then they must master the physics of their choices and make the devices all work together with a real memory that holds actual bits, communicates to the world outside itself, and works off the given power grid. The step from an empty box in an architecture to a design is a major creative step. There is no algorithm for it.

尽管洋基队开发了获胜的建筑，但英国人创造了获胜的设计并因此赢得了比赛。那个盒子的设计，内存，是第一台计算机和数码照片竞争的关键因素。

Although the Yanks developed the winning architecture, the Brits created the winning design and hence won the race. The design of that one box, Memory, was the crucial factor in the race to the first computer—and digital pictures.

冯诺依曼和图灵架构是等效的想法。两者都是实施图灵存储程序通用计算机概念的计划。图灵的架构也有一个存储盒，以及控制、算术、逻辑和输入输出盒。在这两种架构中，内存和输入输出都相当于通用图灵机的磁带，其余的盒子构成了它的扫描头。图灵在 1947 年的一次关于 Ace 的演讲中明确说明了这种联系。同样，尽管 Edvac 的报告并不明确，但冯诺依曼在 1943 年和 1944 年告诉人们图灵的根本重要性。12

The von Neumann and Turing architectures were equivalent ideas. Both were plans for implementing Turing’s stored-program universal computer concept. Turing’s architecture had a box for memory too, and control, arithmetic, logic, and input-output boxes. In both architectures the memory and input-output are equivalent to the tape of the universal Turing machine, and the remaining boxes form its scanning head. Turing made the connection explicit in a lecture on Ace in 1947. Similarly, although the Edvac report wasn’t explicit, von Neumann told people about Turing’s fundamental importance in 1943 and 1944.12

一般的经验法则是，任何软件过程都可以通过在硬件中实现来加快速度。事实上，计算机本身就是图灵的软硬件实现，但速度非常慢，存储程序通用计算机的想法。算术和逻辑可以是软件，但冯诺依曼和图灵走得更远。他们都指定这些非常流行的操作应该在硬件中实现——尽可能快地进行。这是导致人们普遍误解计算机只是算术机的规范。

A general rule of thumb is that any software process can be made to go faster by implementing it in hardware. In fact, a computer itself is a fast hardware realization of Turing’s soft, but painfully slow, idea of a stored-program universal computer. Arithmetic and logic can be software, but von Neumann and Turing went further. They both specified that these very popular operations should be realized in hardware—to go as fast as possible. This is the specification that led to the common misperception that a computer is only an arithmetic machine.

图灵的架构与冯诺依曼团队的架构有着深刻的不同。图灵指定了一个新的硬件支持概念：软件层次结构。程序员（阅读图灵）很快了解到分层创建程序是一种很好的软件实践。层次结构为那些极长的无意义指令的线性列表之一赋予了意义和结构。

Turing’s architecture differed from the von Neumann team’s in a profound way. Turing specified hardware support for a new concept: the software hierarchy. Programmers (read Turing) quickly learned that it’s good software practice to create a program hierarchically. Hierarchy gives meaning and structure to one of those extremely long linear lists of meaningless instructions.

创建软件层次结构就像作者将一本书分成章节列表。假设一个程序有数十万条指令——这在当今世界并不罕见，而且令人难以置信。但是你可以把它分解成一百个部分，每个部分平均有几千条指令，并给每个部分起一个有意义的名字。对程序员来说，程序然后简化为这数百个命名的、有意义的部分——每个都称为一个子程序（不是图灵的话）。在层次结构的最高级别，程序在概念上只有一百条“指令”长，其中每条指令都是一个子程序。它就像一个目录及其章节。

Creating a software hierarchy is like an author breaking a book into a list of chapters. Suppose a program is hundreds of thousands of instructions long—not that unusual in today’s world, and mind-boggling. But you could break it up into, say, a hundred parts averaging a few thousand instructions each, and give each part a meaningful name. To the programmer, the program then reduces to only these hundred named, meaningful parts—each called a subroutine (not Turing’s word). At the highest level of the hierarchy, the program is conceptually only a hundred “instructions” long, where each instruction is a subroutine. It’s like a table of contents and its chapters.

程序员仍然必须创建所有已命名的部分、子程序，以及所有繁琐的细节，而作者仍然必须编写每一章。但是编写每个部分的任务不那么令人难以置信，而且更紧凑。此外，这是一个分层的想法。程序员可以进一步将每个子程序细分为它自己的子子程序，并将它们细分为它们的子程序，依此类推。咬变成小点心，甚至更容易编程。子程序概念将极其复杂的任务驯服为可理解性——例如进一步将章节分解为标题，将标题分解为段落，最后分解为单词本身。图灵的架构指定了一个硬件提升，使子程序的分层使用变得快速和容易。

The programmer still has to create all the named parts, the subroutines, in all their tedious detail, and the author still has to write each chapter. But the task of writing each part is less mind boggling and more bite-sized. Furthermore, it’s a hierarchical idea. A programmer can further subdivide each subroutine into its own sub-subroutines, and they into theirs, and so forth. Bites become nibbles, even easier to program. The subroutine concept tames an overwhelmingly complex task into comprehensibility—like further breaking chapters into headings, headings into paragraphs, and finally into the words themselves. And Turing’s architecture specified a hardware boost that makes the hierarchical use of subroutines quick and easy.

当然，图灵的架构——具有复杂的子程序支持——应该是被采用的架构，但事实并非如此。Yanks 在 1940 年代广泛传播冯诺依曼的 Edvac 报告，但英国人直到 1986 年才公开发表 Turing 的 Ace 报告。13

Surely Turing’s architecture—with its sophisticated subroutine support—should’ve been the one that got adopted, but it wasn’t. The Yanks disseminated von Neumann’s Edvac report far and wide in the 1940s, but the Brits didn’t publish Turing’s Ace report publicly until 1986.13

英国人

The Brits

洋基队

The Yanks

数码灯

Digital Light

英国人也在与洋基队争夺快速存储设备。对于 Digital Light 来说幸运的是，两个团队都在寻找可见的真空管记忆。两者都将位存储为显示在二维网格上的不同大小的点。因此，每个位置自然既是一位扩展像素又是一位存储器。英国人通过改编普通阴极射线管——威廉姆斯管赢得了这场小规模冲突，然后是这场战斗。洋基队也立即采用了该解决方案。

The Brits vied with the Yanks in the race for a fast memory device too. Luckily for Digital Light, both teams looked for visible vacuum tube memories. Both would store bits as spots of different sizes displayed on a two-dimensional grid. Hence each location was naturally both a one-bit spread pixel and a one-bit memory. The Brits won this skirmish, and then the battle, with an adaptation of an ordinary cathode-ray tube—the Williams tube. The Yanks immediately adopted that solution too.

奇怪的耳饰

Weird Eadweard

他起初是泰迪马格里奇。或者也许是艾迪。我们不知道他妈妈真正的喜爱，但直到很久以后他肯定不是 Eadweard Muybridge。这个名字是他众多别名中的最后一个，也是最奇怪的一个。他的正式名字是爱德华·詹姆斯·马格里奇，出生在伦敦附近的泰晤士河畔金斯敦。这张表记录了这位著名表演者在公众面前进行的名字游戏——他的名字的变体与他的姓氏的变体一起使用：3

He was Teddy Muggeridge at first. Or perhaps Eddy. We don’t know his mum’s actual endearment, but he certainly wasn’t Eadweard Muybridge until much later. That name was the last—and the weirdest—of his many aliases. Officially he was Edward James Muggeridge, born in Kingston upon Thames near London. This table captures the name game that this notable showman worked on the public—with variations of his first name used with variations of his surname:3

他在这里永远是 Edward Muybridge（简称为“我的桥”），因为他在这个别名下完成了他最伟大的工作——对 Digital Light 很重要的工作。

He’ll always be Edward Muybridge (pronounced simply “My bridge”) here because he did his greatest work—the work that matters to Digital Light—under that alias.

他以 EJ Muggridge 的身份来到美国——带着一个gg。到1860 年，当他登上一辆从加利福尼亚开往东部的巴特菲尔德陆路驿马车时，他已成为 EJ Muygridge（拥有yg ）。他没有成功。对他的未来如此重要的横贯大陆铁路将是一个更好的选择——但它还不存在。事实上，六匹巴特菲尔德马在德克萨斯州的某处沿着著名的舞台线吓坏了，他被从马车上扔了下来。他遭受了足够的身体——也许是精神——的伤害，以至于起诉巴特菲尔德并赢得了和解。1861 年，他仍然是 Muygridge，现在回到英国，当时他试图为洗衣机申请专利。他是第一个迈布里奇——与yb——1862 年在另一项英国专利申请中，这一项用于印刷工艺。奇怪的是，他从未尝试为他最重要的发明——Zoöpraxiscope——申请专利，而这项发明尚未到来。（电影设备的名字和电脑的名字一样有趣。）

He had come to America as E. J. Muggridge—with a gg. He had become E. J. Muygridge—with a yg—by 1860 when he boarded a Butterfield Overland stagecoach from California bound for the East. He didn’t make it. The transcontinental railroad that was so important to his future would’ve been a far better choice—but it didn’t exist yet. As it was, the six Butterfield horses spooked somewhere in Texas along the famous staged line, and he was thrown from the coach. He suffered sufficient bodily—and perhaps mental—damage to sue Butterfield and win a settlement. He was still Muygridge in 1861, now returned to England, when he tried to patent a washing machine. He was first Muybridge—with the yb—in 1862 in another English patent application, this one for a printing process. Oddly, he never tried to patent his most important invention—the Zoöpraxiscope—which was yet to come. (The names of movie devices are as much fun as those of computers.)

他最丰富多彩的别名是 Helios（只有一个字）。在约翰·缪尔（John Muir）首次支持加利福尼亚奇妙的优胜美地山谷（Yosemite Valley）时，他开始成为一种新的艺术家，摄影艺术家，在旧金山取得了这个名字。赫利俄斯（太阳神摄影师）拍摄了一些最早且广受好评的强大优胜美地照片。

His most colorful alias was Helios (just the one word). He assumed the name in San Francisco as he set out to be a new kind of artist, a photographic artist, near the time when John Muir first championed California’s marvelous Yosemite Valley. Helios—the photographer as sun god—made some of the earliest, and acclaimed, photographs of mighty Yosemite.

正是他的谋杀案审判让他声名狼藉——又是迈布里奇先生，现在是爱德华迈布里奇。一个全是男性的陪审团裁定他在旧金山附近枪杀了他妻子的情人——以及他孩子可能的父亲——无罪。毫无疑问，他杀死了这个人——用一把他完全打算用于此目的的手枪直指胸膛。尽管有法官的告诫，陪审团并不打算在加州淘金热之后找到一个捍卫自己“荣誉”（！？）的人犯有谋杀罪。4

It was his murder trial that brought him notoriety—as Mr. Muybridge again, now Edward Muybridge. An all-male jury acquitted him of gunning down his wife’s lover—and his child’s probable father—near San Francisco. There’s no question that he killed the man—point blank to the chest with a pistol that he fully intended for the purpose. Despite the judge’s admonition, the jury wasn’t about to find a man defending his “honor” (!?) guilty of murder in post–Gold Rush California.4

迈布里奇只是在晚年 52 岁时才影响了古怪的 Eadweard，因为他在巴黎的艺术界声名鹊起。他肯定是从家乡泰晤士河畔金斯敦著名的加冕石（图 5.1）中得到的。900 年代，两位名叫爱德华的国王在这块石头上加冕。在它的底部两次浮雕是Eadweard - 古英语拼写。所以迈布里奇的最终名字只是“Ed ward”，尽管转录很容易起皱。

Muybridge affected the weird Eadweard only later in life, at age 52, as he made a name for himself in the art world of Paris. He took it surely from the famous coronation stone (figure 5.1) in his hometown, Kingston upon Thames. Two kings named Edward had been crowned on that stone in the 900s. Embossed in its base twice is Eadweard—the Old English spelling. So Muybridge’s final first name was just “Ed ward,” despite the pucker-provoking transcription.

在 Helios 和 Eadweard 之间，普通的爱德华完成了他令人难忘的工作。他的赞助人利兰·斯坦福（Leland Stanford）从建造第一条横贯大陆的铁路中获得了令人发指的强盗大亨财富——这些铁马将巴特菲尔德驿马车及其多肉的马带入了遗忘（或修剪整齐的椭圆形轨道）。当然，他巨额财富的主要遗产是斯坦福大学，硅谷将在斯坦福大学（在 1960 年代）成为摩尔定律的温床。但在他自己的时代，斯坦福大学在赛马——他的热情——尤其是猪身上投入了大量资金。

Between Helios and Eadweard, plain Edward did his memorable work. His patron, Leland Stanford, gained outrageous, robber-baron wealth from construction of the first transcontinental railroad—the iron horses that railroaded the Butterfield stagecoach and its fleshy horses into oblivion (or onto a manicured oval track). The major legacy of his vast fortune was, of course, Stanford University, from which Silicon Valley would spring (in the 1960s) as a hotbed of Moore’s Law. But in his own time, Stanford spent heavily on racehorses—his passion—particularly trotters.

爱德华·迈布里奇 (Edward Muybridge) 以这个名字着称，他在中途小跑时拦住了斯坦福最喜欢的小跑之一，一张照片显示所有四只蹄子都离地。公认的智慧是迈布里奇从那里开始发明电影。这归功于他第一次演示，当以相似的速度投影时，一系列快速连续拍摄的运动物体（例如马或裸体男人或女人）的静止照片似乎显示了运动中的物体。但这个故事并非如此。5

Edward Muybridge, under that name, famously stopped one of Stanford’s favorite trotters in mid-trot with a single photograph showing all four hooves off the ground. Received wisdom is that Muybridge proceeded from there to invent movies. It credits him with the first demonstration that a sequence of still photographs of a moving object, such as a horse—or a naked man or woman—taken in quick succession appears to show the object in motion, when projected at similar speed. But this story isn’t quite so.5

他实际上所做的是这样的。他在斯坦福的农场上建造了一个 24 摄像头的设备——The Farm，这是斯坦福学生今天对校园的称呼。他使用了一个巧妙的触发机制，当一匹马疾驰而过时，快速连续地打开每个相机的快门。每个照相机在涂有化学乳剂的玻璃板上形成一个单一的摄影图像。这些印版必须立即在现场开发。6

What he actually did was this. He built a 24-camera apparatus on Stanford’s farm—The Farm, as Stanford students call the campus today. He used a clever triggering mechanism to open the shutter of each camera in quick succession as a horse raced by. Each camera formed a single photographic image on a glass plate coated with a chemical emulsion. The plates had to be developed immediately on site.6

为了展示结果，他使用了公共演讲者用来投射静止图像的著名魔术灯——当时的 PowerPoint。他在其中添加了一个比旧的长播放 (LP) 唱片更大的旋转磁盘。他在圆盘的边缘放置了小画，每幅画都来自连续的马照片，如图 5.2 所示（左，快门在右）。当圆盘以或多或少均匀的速度旋转时，投影图像似乎在移动，再现了马的运动。一个完整的循环大约需要一两秒。7

To show the results he took the well-known magic lantern that public speakers used to project still images—the PowerPoint of its time. He added to it a rotating disk larger than an old long-playing (LP) record. He placed small paintings, each derived from successive horse photographs, around the edge of the disk, as shown in figure 5.2 (left, with the shutter at right). As the disk rotated at a more-or-less uniform speed, the projected images appeared to move, recreating the movement of the horse. One complete cycle took about one or two seconds.7

1880 年 1 月 16 日，迈布里奇在 Leland Stanford 位于旧金山的豪宅中放映了这部假定的第一部电影。但这种说法存在几个问题。为了例如，他没有投影相框。它们是从照片中衍生出来的画作。他用一定程度的扭曲修改了画作——水平拉长的腿和身体——以适应他的投影系统的扭曲伪影。而且他没有电影摄影机，只有一系列静态摄影机，每个摄影机的视角略有不同。最后，只有他可以使用该系统，因为它已经固定到位。说一个没有电影摄影机、不使用胶卷、不投影照片的系统是一个电影系统，这是一个太大的飞跃。就像几乎是计算机一样，他的系统充其量也就是电影，这是一个延伸。8

Muybridge showed this supposed first movie in Leland Stanford’s San Francisco mansion on January 16, 1880. But there are several problems with this claim. For instance, he didn’t project photographic frames. They were paintings derived from photographs. And he modified the paintings with a certain amount of distortion—horizontally elongated legs and bodies—to accommodate a distortion artifact of his projection system. And he didn’t have a movie camera, only a sequence of still cameras, each with a slightly different point of view. And, finally, only he could use the system, as it was fixed in place. It’s too large a leap to say that a system that didn’t have a motion-picture camera, didn’t use film, and didn’t project photographs was a movie system. Like the almost-computer, his system was at best almost-cinema, and that’s a stretch.8

迈布里奇并不是第一个展示静止图像运动的人，无论是摄影静止图像还是从中衍生出的轮廓。在 1860 年代和 1870 年代流行的 Zoëtrope 做到了这一点（图 5.3）。想象一个大约半英尺高、直径两英尺的纸板圆筒。如果你打开盒子里的灯，如果你足够小，可以放进盒子里，你会看到围绕圆柱体圆周的一系列图片，它们的顶部对齐。每张图片都是一个序列中的一张，比方说一匹马在小跑中连续走几步。当然，你不能装在圆柱体里面，所以为了让你从外面窥视，每张照片之间的墙壁上都开有缝隙。通过每个狭缝，您可以看到对面墙上的一张图片。如果你要在圆柱体外面走来走去，你会看到每个缝隙都显示下一张图片，即马小跑的下一部分。现在想象一下，当圆柱体转动时，您从外部固定位置观察圆柱体。在狭缝之间，你看到了黑暗。只有当狭缝旋转到位时，您才能看到图片。就是正对面的图片，是马的下一段小跑。因此，当圆柱体以适当的速度旋转时，您会看到一系列照片，显然都在同一个位置，从您的眼睛穿过圆柱体。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。现在想象一下，当圆柱体转动时，您从外部固定位置观察圆柱体。在狭缝之间，你看到了黑暗。只有当狭缝旋转到位时，您才能看到图片。就是正对面的图片，是马的下一段小跑。因此，当圆柱体以适当的速度旋转时，您会看到一系列照片，显然都在同一个位置，从您的眼睛穿过圆柱体。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。现在想象一下，当圆柱体转动时，您从外部固定位置观察圆柱体。在狭缝之间，你看到了黑暗。只有当狭缝旋转到位时，您才能看到图片。就是正对面的图片，是马的下一段小跑。因此，当圆柱体以适当的速度旋转时，您会看到一系列照片，显然都在同一个位置，从您的眼睛穿过圆柱体。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。只有当狭缝旋转到位时，您才能看到图片。就是正对面的图片，是马的下一段小跑。因此，当圆柱体以适当的速度旋转时，您会看到一系列照片，显然都在同一个位置，从您的眼睛穿过圆柱体。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。只有当狭缝旋转到位时，您才能看到图片。就是正对面的图片，是马的下一段小跑。因此，当圆柱体以适当的速度旋转时，您会看到一系列照片，显然都在同一个位置，从您的眼睛穿过圆柱体。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。从这一系列样本中，您可以看到一匹马在小跑。您可能已经意识到这或多或少也是电影电影的工作方式，鼓的黑暗部分用作快门。图 5.3 由 William Kennedy Laurie (WKL) Dickson 绘制——这个名字值得记住。9

And Muybridge wasn’t anywhere near the first to show movement from stills, either photographic stills or silhouettes derived from them. The Zoëtrope, popular in the 1860s and 1870s did that (figure 5.3). Imagine a cardboard cylinder about a half foot high and two feet in diameter. If you turned on a light inside the box, and if you were small enough to fit inside, you would see a sequence of pictures around the circumference of the cylinder, their tops aligned. Each picture would be one from a sequence, let’s say of a horse taking successive steps during a trot. Of course, you can’t fit inside the cylinder, so to allow you to peer in from the outside, slits are cut in the walls between each picture. Through each slit you see one picture positioned on the opposite wall. If you were to walk around outside of the cylinder, you would see that each slit reveals the next picture, the next part of the horse’s trot. Now imagine that you view the cylinder from a fixed position outside as it turns. Between the slits, you see darkness. You see a picture only when a slit rotates into position. It’s the picture directly opposite, which is the next part of the horse’s trot. So, as the cylinder rotates at the proper speed you see a succession of pictures, all apparently in the same location, across the cylinder from your eye. From this succession of samples, you perceive a horse moving in a trot. You probably recognize that this is more-or-less how film movies work, too, with the dark parts of the drum serving as the shutter. Figure 5.3 was drawn by William Kennedy Laurie (W.K.L.) Dickson—a name worth remembering.9

Zoëtrope 中的图片可以是照片或源自照片的绘画。如果你忽略图像来源于照片的要求，那么中国人在大约两千年前就发明了西洋镜。西洋镜中的图像不仅限于马。10

The pictures in a Zoëtrope could be photographs or paintings derived from photographs. If you omit the requirement that the images originate as photographs, then the Chinese invented the Zoëtrope almost two millennia ago. And the images in a Zoëtrope weren’t restricted to horses.10

迈布里奇也不是第一个投影图像序列的人。Athanasius Kircher 在 1671 年就这样做了，或者至少写过它。关于写作的附带条件是必要的，因为 Kircher 将镜头放在幻灯片的错误一侧（图 5.4），如果他实际制造了投影仪，他会注意到这一点。实施是橡胶与创意之路相遇的地方。11

Nor was Muybridge the first to project sequences of images. Athanasius Kircher was doing that in 1671, or at least writing about it. The proviso about writing is necessary because Kircher placed the lens on the wrong side of the slides (figure 5.4), something he would’ve noticed had he actually built the projector. Implementation is where the rubber meets the road for ideas.11

在本章的后面，我们将从现实世界中拍摄的真人电影到由虚幻世界制作的动画电影。令人惊讶的是，迈布里奇的原创贡献与动画电影更为相关。上个世纪的动画师通过用墨水逐帧追踪在摄影胶片上运动中捕捉到的人或动物，从电影中衍生出卡通片。这种技术，称为rotoscoping，是迈布里奇为他所谓的第一部电影所做的。

Later in this chapter we’ll proceed from live-action films taken in the real world to animated movies made of unreal worlds. Muybridge’s original contributions are more pertinent, surprisingly, to animated movies. Animators of the last century derived cartoons from film by tracing in ink, frame by frame, over humans or animals captured in motion on photographic film. This technique, called rotoscoping, is what Muybridge did for his supposed first movie.

动画师也以扭曲他们的卡通而闻名——使用称为挤压和拉伸、夸张和预期的技术——以适应低幼稚的采样电影率。迈布里奇在那部所谓的第一部电影中拉伸了马匹（或为他完成了工作）。仔细查看迈布里奇光盘上的图像。因此，他可以更恰当地被人们铭记为第一个使用转描和拉伸的人，这一成就有点羞于发明电影。12

Animators also famously distort their cartoons—with techniques called squash and stretch, exaggeration and anticipation—to accommodate for the low naive sampling rate of movies. Muybridge stretched the horses in that supposed first movie (or had the work done for him). Look carefully at the images on the Muybridge disc. So he can more properly be remembered as the first to use rotoscoping and stretch, an accomplishment a bit shy of inventing cinema.12

如果你想一想，迈布里奇最真正令人难忘的遗产是那些移动的裸体和动物的连续照片的奇妙静态阵列，今天仍然普遍作为他的作品集的明信片、海报或专辑。它们像数字图像中的像素光栅一样从左到右和从上到下排列。

If you think about it, Muybridge’s most truly memorable legacy are those marvelous static arrays of sequential photographs of moving nudes and animals, still pervasive today as postcards, posters, or albums of his collected works. They are arrayed left to right and top to bottom like the raster of pixels in a digital image.

动画师高度重视这些迈布里奇光栅。在 1920 年代初期，两个年轻的朋友，Ub Iwerks 和 Walt Disney，在他们推出了他们刚刚起步的动画业务时向他们学习。动画师今天仍然尊重他们。图 5.5 是 Pixar 动画师的恶搞和高度赞扬。迈布里奇为斯坦福大学著名地证明了一匹马的所有四英尺都在一帧内离开地面，而这项“最近重新发现的工作”确定了 Luxo 台灯的一英尺离开地面四帧。

Animators hold these Muybridge rasters in high regard. In the early 1920s two young pals, Ub Iwerks and Walt Disney, learned from them as they launched their fledgling animation business. And animators still honor them today. Figure 5.5 is a spoof—and high tribute—by Pixar animators. Muybridge famously proved for Stanford that all four feet of a horse leave the ground in one frame, while this “recently rediscovered work” establishes that the one foot of a Luxo desk lamp leaves the ground for four frames.

一些作家指责强盗男爵斯坦福从迈布里奇那里窃取信用，但检查记录表明并非如此。几十年前，在浏览在一家古董书店，我发现了一本 1882 年的大型毛绒书，名为《奔跑的马》。我相信这是最初的迈布里奇研究，对任何对动画感兴趣的人都很有名。

Some writers have accused robber-baron Stanford of stealing credit from Muybridge, but inspection of the record indicates otherwise. Several decades ago, while browsing an antique bookstore, I discovered a large plush book from 1882 titled The Horse in Motion. I leapt at it believing it to be the original Muybridge study that was famous to anyone interested in animation.

但事实并非如此——反正也不完全是。首先，作者不是迈布里奇。是我从未听说过的 JDB Stillman。打开扉页，我发现 Leland Stanford 确实资助了该出版物，正如预期的那样。迈布里奇在哪里？我很快就找到了他，作为“先生。迈布里奇，”在第一段中突出显示。“由 EJ Muybridge 先生提供”的附录描述了他的技术。和“先生。迈布里奇”也出现在最后一段，从而包含了整本书。13

But it wasn’t—not exactly anyway. For starters, the author wasn’t Muybridge. It was J. D. B. Stillman, whom I’d never heard of. Opening to the title page, I found that Leland Stanford had indeed funded the publication, as expected. Where was Muybridge? I soon found him, as “Mr. Muybridge,” prominently displayed in the very first paragraph. An appendix “furnished by Mr. E. J. Muybridge” described his technique. And “Mr. Muybridge” appeared in the very last paragraph too, thus embracing the entire book.13

从正文中可以清楚地看出，斯坦福和斯蒂尔曼根本不关心摄影。马的运动对他们来说很重要。迈布里奇的照片与其说是艺术品，不如说是出租作品——称它们为意味着结束。斯蒂尔曼确实提到了作品的艺术意义，但他的意思只是艺术家对马的准确演绎。马就是问题。那么，为什么迈布里奇会因为这本书起诉斯坦福？

It’s clear from the main text that Stanford and Stillman simply didn’t care about photography. The movement of horses was the important thing to them. Muybridge’s photographs served less as works of art than they did as works for hire—call them a means to an end. Stillman does mention the artistic import of the work, but he meant only the accurate rendition of horses by artists. The horses were the thing. So why did Muybridge sue Stanford over this book?

到其出版时，他已前往巴黎并为自己加冕为高贵的 Eadweard（图 5.6，右）。他向那里的艺术界宣布，他发明了一种新的艺术形式，并且是大亨斯坦福的好友（图 5.6，左）。他用马和人在运动中让他们惊叹不已。他是预测运动的雄狮营销者，他受到了崇拜。他还说服了艾萨克·牛顿在英国的皇家学会接受他的论文《运动中动物的态度》并安排出版。但随后斯蒂尔曼的书出现了，仅仅提到了受雇工作，斯坦福声称拍摄一匹运动的马是他自己的想法。重要的是，迈布里奇的名字不在扉页上。皇家学会拒绝发表这篇论文，怀疑是抄袭。迈布里奇被羞辱了。他的一些新粉丝认为他是个骗子。14

By the time of its publication he had gone to Paris and crowned himself the noble Eadweard (figure 5.6, right). He announced to the artistic community there that he had invented a new artform and was a buddy of tycoon Stanford (figure 5.6, left). He wowed them with horses and men in motion. He was a leonine marketer of projected movement, and he was lionized. He also had convinced Isaac Newton’s Royal Society in England to accept his paper On the Attitudes of Animals in Motion and schedule it for publication. But then Stillman’s book appeared with the mere work-for-hire mention and Stanford’s claim that it was his own idea to film a horse in motion. Importantly, Muybridge’s name wasn’t on the title page. The Royal Society reneged on the offer to publish the paper, suspecting plagiarism. Muybridge was humiliated. Some of his new fans thought he was a fraud.14

他把西装弄丢了。现在这一切似乎都被误导了。斯坦福的一切都是关于马的。迈布里奇是关于摄影技术的。拍摄运动中的马是斯坦福在 1872年的想法——这一想法受到 1874 年科普文章的重新启发关于 Étienne-Jules Marey 在法国的工作。但正是迈布里奇的发明使之成为可能。这是一场不同形式的塔楼与恶臭之战——理念与工程。斯坦福将这些照片全部归功于他为这项任务聘请的人。事实上，他以一美元的价格将这项技术的知识产权转让给了迈布里奇。他完全错过了这项技术的潜在进口，无论是在艺术上还是在经济上。15

And he lost the suit. It all seems so misguided now. Stanford was all about the horses. Muybridge was all about the photographic technique. It was Stanford’s idea in 1872 to film the horse in motion—an idea re-inspired by an 1874 Popular Science article about the work of Étienne-Jules Marey in France. But it was Muybridge’s invention that made it possible. It’s a tower versus stinks battle in a different guise—the idea versus the engineering. Stanford gave full credit for the photographs—to the man he had hired for the task. In fact, he signed away the intellectual property rights for the technique to Muybridge—for a dollar. He completely missed the potential import of the technique, both artistically and financially.15

然而，富有的斯坦福大学凭借知名度、金钱、设备和空间，让发明家迈布里奇向世界展示了快速拍摄以相同快速序列投影的连续照片给人一种运动的错觉。斯坦福为此付出了代价——付出了很多——却从来不知道他到底资助了什么。这不是关于马的。华丽的迈布里奇成功地将这个想法推向了广泛的领域——这也许是他对电影最重要的贡献。“电影之父”对他来说似乎离谱，但“电影的约翰尼苹果种子”却很合适。特别是，他的节目激发了托马斯·阿尔瓦·爱迪生对美国电影的兴趣，并影响了法国的马雷。这就是即将到来的洋基队与法兰克队比赛的第一缕曙光。16

Nevertheless, the wealthy Stanford made it possible—with name recognition, money, equipment, and space—for Muybridge the inventor to show the world that rapidly shot sequential photographs projected in the same quick sequence give the illusion of motion. Stanford paid for it—paid a lot—without ever understanding exactly what it was that he had funded. It wasn’t about the horses. And flamboyant Muybridge successfully marketed the idea far and wide—perhaps his most important contribution to cinema. “The father of cinema” seems way off the mark for him, but “the Johnny Appleseed of cinema” fits nicely. In particular, his show inspired Thomas Alva Edison’s interest in motion pictures in America and influenced Marey in France. And that’s the first glimmer of the Yanks versus Franks contest to come.16

定义电影

Defining a Movie

对计算机的仔细定义帮助我们修剪了计算机故事中的历史主张。它扫清了通往 Digital Light 的道路。我们可以对电影史采用相同的技术。

A careful definition of computer helped us prune the thicket of historical claims in the story of computers. It cleared the path to Digital Light. We can employ the same technique for cinema history.

让我们也使用另一种适用于计算机的方法：将硬件与软件分开。对于电影院来说，电影本身就是软件。事实上，软件本身就是我们通常所说的“电影院”。我们可以在电影、电视或手机上看到《公民凯恩》。像往常一样，神奇之处在于软件——钢琴中的肖邦练习曲。但这本书的一个基本概念建立在对我们现代媒体文化的欣赏之上，而现代媒体文化本身依赖于对其技术基础的直观理解——在这两个领域中看到奇迹和美景。由于有大量关于电影软件（电影）的书籍，我们专注于未被充分认识的硬件。

Let’s also use another approach that worked for computers: separate hardware from software. For cinema, the movies themselves are the software. In fact, the software alone is what we usually mean by “the cinema.” We can see Citizen Kane on film, television, or cellphone. As usual, the magic lies in the software—the Chopin étude in a piano. But a fundamental notion of this book rests on an appreciation of our modern media culture that itself depends on an intuitive understanding of its technological underpinnings—on seeing marvels and beauties in both realms. Since there are a host of books on movie software—the cinema—we concentrate on the underappreciated hardware.

但在过于仓促地转向硬件之前，让我们先看看电影软件的一个奇怪之处。尽管电影放映机只能沿电影的一个方向移动，但存储在电影上的软件可以让观众感知时间倒退（倒叙，爱尔兰人）或向前（比如十年后，甚至千年，2001：太空漫游）的事件），或者无限循环（土拨鼠日），或者随着变化的重复（罗生门）。就好像电影软件在我们的大脑上计算了它在电影硬件上无法计算的东西。

But before heading too hastily toward hardware, let’s notice a curious thing about cinema software. Although a movie projector moves along film in one direction only, software stored on the film can make viewers perceive events that jump backward in time (flashbacks, The Irishman) or forward (say ten years later, or millennia even, 2001: A Space Odyssey), or in infinite loops (Groundhog Day), or as repetition with change (Rashomon). It’s as if movie software computes on our brains what it can’t compute on movie hardware.

没有一个词可以形容那种硬件，电影院的硬件。经典电影具有三个可分离的部分：相机、胶卷和投影仪。通过说电影机器，我们指的是所有三个——就好像它们在一个黑盒子里一样——这就是我们的定义。电影机器之于电影就像计算机之于计算。少了任何东西，充其量就是一台电影机器——就像上一章的电脑一样。由于硬件在这里是隐含的，所以我经常放弃机器这个词，并将电影和电影这两个术语与电影互换使用。17

There’s no single word for that hardware, the hardware of cinema. Classic cinema has three separable pieces: camera, film, and projector. By saying movie machine, we mean all three—as if they were in one black box—and that’s our definition. A movie machine is for cinema what a computer is for computation. Anything less is, at best, an almost movie machine—like the almost-computer of the last chapter. Since hardware is implicit here, I often drop the word machine and use the terms motion picture and cinema interchangeably with movie.17

所以，当我们问谁制作了第一部电影时，我们并不是指第一个内容。我们指的是谁拥有第一台电影机——第一台相机、胶片和投影仪作为一个系统协同工作。这就是上世纪大部分时间我们所说的电影，硬件方面的意思，今天仍然如此。在数码光中，我们有数码相机，存储数码相框，并用数码投影仪进行投影。

So, when we ask who made the first movie, we don’t mean the first content. We mean who had the first movie machine—the first camera, film, and projector working together as one system. That’s what we meant by a movie, hardware-wise, most of last century, and it still means that today. In Digital Light, we have digital cameras, store digital frames, and project them with digital projectors.

为什么电影有效？

Why Do Movies Work?

电影应该起作用并不明显。电影并不像进入我们眼睛的视觉那样流畅。相机每秒只记录 24 个视觉流的快照，并丢弃帧之间发生的一切——但我们还是感知到了流。我们看到静止，但我们感知到运动。我们该如何解释呢？

It’s not obvious that movies should work. Movies are not smooth like the visual flow into our eyes. The camera records only 24 snapshots of each second of visual flow, and discards everything that happens between frames—but we perceive flow anyway. We see stills, but we perceive motion. How can we explain this?

早期影院系统流程图

A Flow Chart of Early Cinema Systems

与计算机一样，需要一个复杂的流程图才能开始代表电影硬件的早期历史。它也让我们看到了人、概念和机器之间的复杂关系。这也是对实际故事的简化。许多名称、设备和国家被省略以减少混淆程度，同时保留足够的细节来表明重要技术开发的实际复杂性。与计算机一样，许多其他播放器和机器出现在注释中。

As with computers, it takes a complicated flow chart to even begin to represent the early history of movie hardware. It too allows us to see the complicated relationships of people, concepts, and machines. It too is a simplification of the actual story. Many names, devices, and countries are omitted to reduce the confusion level, while enough detail remains to signal the actual complexity of the development of an important technology. As with computers, many of the other players and machines appear in the annotations.

上一章中计算机历史的简化流程图仅保留了最终对 Digital Light 重要的人员和发展。这里的等效简化（图 5.10）是只遵循那些类似于前千禧年时期（几乎正是二十世纪（1895-2000））的主导电影系统的系统。我们将那些无胶片或无框架的系统降级为“电影前”——任何基于简单旋转的 Zoëtrope 或 Zoöpraxiscope。

The simplified flow chart of computer history in the previous chapter retained only those people and developments that mattered eventually to Digital Light. The equivalent simplification here (figure 5.10) is to follow only those systems that resemble the dominant movie system of the pre-millennial period—almost exactly the twentieth century (1895–2000). We relegate to “pre-cinema” those systems that were filmless or frameless—anything like the Zoëtrope or Zoöpraxiscope based on simple rotation.

一个主要的简化是从这张图表中完全省略了光化学薄膜本身。提到的每个系统都依赖于它。掌握赛璐珞电影对早期电影系统至关重要。这是一个困难的媒介。它很脆，容易划伤，边缘卷曲，并且具有化学活性。发明者必须制作曝光时间短、分辨率高的胶片。开发和打印它们必须快速且可靠。对于早期的发明者来说，这是一场噩梦，但对于用比特代替胶片的 Digital Light 来说，这并不重要。24

A major simplification is the complete omission of photo-chemical film itself from this chart. Every system mentioned depended on it. Mastery of celluloid film was crucial to early movie systems. It was a difficult medium. It was brittle, scratched easily, rolled at the edges, and was chemically active. Inventors had to create film stocks with short exposure times and high resolution. Developing and printing them had to be rapid and reliable. It was a nightmare for the early inventors, but it matters not a whit for Digital Light, which substitutes bits for film.24

再一次，与计算机一样，我们调用奥林匹克隐喻来跟踪许多球员。这次的主要球队是洋基队和法兰克队。为这张流程图选择的玩家促成了五个强大工作室的崛起，Edison、Biograph、卢米埃尔、高蒙和百代。一张真正显示对早期电影的所有贡献的详细图表必然会显示许多其他早期发明者和设备，并且它将至少包含另外两个国家，一个英国团队和一个德国团队。25

And again, as with computers, we invoke an Olympic metaphor to track the many players. The principal teams this time are the Yanks and the Franks. The players selected for this flow chart contributed to the rise of five powerhouse studios, Edison, Biograph, Lumière, Gaumont, and Pathé. A detailed chart that really showed all contributions to early cinema would necessarily show a host of other early inventors and devices, and it would feature at least two other countries, a Brits team and a German team.25

也许最有趣的遗漏是法国、英国和美国的路易·勒王子。1888 年，他在约克郡的利兹制作了短片。它们仍然可以在互联网上以数字重建的形式找到。1890 年，他在法国登上了一列火车，显然是前往巴黎，最终前往纽约展示他的发明。但他从未成功——甚至没有到巴黎。他只是消失了。他的家人认为爱​​迪生将他视为一个严肃的竞争对手，并将他除名——这不太可能发生，但却表明了爱迪生的暴虐名声。根据一份宣誓书，到 1889 年，勒普林斯拥有了第一个完整的摄影机、胶卷和放映机系统。但这并不重要，因为它显然对电影的发展没有进一步的影响。26

Perhaps the most intriguing omission is Louis Le Prince of France, England, and the United States. He made short movies in Leeds, Yorkshire, in 1888. They can still be found on the internet as digital reconstructions. In 1890 he boarded a train in France, apparently headed for Paris and ultimately New York to demonstrate his invention. But he never made it—not even to Paris. He simply disappeared. His family believed that Edison saw him as a serious competitor and had him removed—an unlikely event but indicative of Edison’s tyrannical reputation. According to an affidavit, Le Prince had the first complete system of camera, film, and projector by 1889. But it doesn’t matter because it apparently had no further effect on the development of cinema.26

下面的文字作为电影流程图的扩展说明——首先是洋基队，然后是法兰克队。但它最重要的信息是明确的：它所代表的历史是复杂的，有着压倒性的细节。推论出来的信息是，如果我们听到有人声称自己是电影之父或发明者，我们应该立即发出危险信号。第二个主要信息是 1895 年是电影年，就像 1948 年是计算机年一样。1895 年是放映机的一年，也是第一次使用赛璐珞胶片实现完整电影机器的一年。

The text that follows serves as an extended caption for the flow chart of cinema—first for the Yanks side and then for the Franks. But its overriding message is clear: the history it represents is complex, with overwhelming details. The corollary message is that we should immediately run up a red flag if we hear one name claimed as father of, or inventor of, cinema. The second principal message is that 1895 was The Year for movies, the way that 1948 was The Year for computers. And 1895 was The Year for the projector, and of the first realizations of full movie machines using strips of celluloid film.

洋基队

The Yanks

法兰克人

The Franks

电影史流程图的右侧是另一个团队——法兰克人。一眼就能看出主要参与者之间的紧密联系网络。他们的故事与 Yanks 的故事一样复杂、强大和令人担忧。我们从 Étienne-Jules Marey 开始。他占据了爱德华·迈布里奇为洋基队担任的弗兰克斯队的鼓舞人心的电影前职位。事实上，马雷和迈布里奇在 1895 年投影仪年之前的几年里相互影响。

The right side of the flow chart of cinema history is devoted to the other team—the Franks. A glance at it reveals a dense web of interconnectedness among the main players. Their story is as complicated, robust, and fraught as the Yanks story. We start at the top with Étienne-Jules Marey. He occupies the inspirational, pre-cinema position for the Franks held by Edward Muybridge for the Yanks. And in fact, Marey and Muybridge influenced one another in the years leading up to 1895, the Year of the Projector.

爱迪生信托与好莱坞的创造

The Edison Trust and the Creation of Hollywood

到 1907 年，爱迪生公司拥有美国电影摄影机的一项主要专利，并通过不断的诉讼来行使这项专利。终于筋疲力尽，爱迪生的竞争对手向公司寻求救济。解决方案是 1908 年电影专利公司的诞生，有时也称为爱迪生信托公司。大约 10 家公司将他们的专利集中在一起，并作为一个团队进行竞争。该集团包括法国公司 Pathé 的美国分公司和当时最大的胶卷供应商伊士曼柯达公司。甚至爱迪生最大的竞争对手迪克森的传记也加入了。爱迪生无法抗拒的传记投影仪专利是使信托如此有效的原因。如果这听起来像垄断勾结，那是因为它是。66

By 1907 the Edison company owned one of the major US patents on movie cameras and exercised it with incessant litigation. Finally exhausted, Edison’s competitors approached the company for relief. The solution was the birth in 1908 of the Motion Picture Patents Company, sometimes called the Edison Trust. About 10 companies pooled their patents together and as a group went after competition. Included in the group was the American branch of the French company, Pathé, and the Eastman Kodak company, the largest supplier of film stock at the time. Even Dickson’s Biograph, Edison’s greatest competitor, joined. The Biograph projector patents—which Edison couldn’t combat—were what made the Trust so effective. If this sounds like monopolistic collusion, that’s because it was.66

声称爱迪生信托导致好莱坞诞生的说法过于强烈。这似乎是合理的，因为加利福尼亚距离总部位于新泽西的爱迪生公司还有很长的路要走。毫无疑问，信托基金发现很难在这种情况下追踪和行使专利。但更重要的西海岸福利是充足的阳光、廉价的劳动力和房地产。许多独立电影制片人搬到了那里。例子有环球影城，它在 1912 年成立后不久就搬到了好莱坞，还有派拉蒙影业，于 1912 年在那里成立，它们今天仍然在好莱坞。这两个 Yanks 的制片厂加上 Gaumont 和 Pathé for the Franks 是世界上现存的五个最古老的电影制片厂中的四个。（第五是丹麦的诺德电影。）

A claim that the Edison Trust led to the creation of Hollywood is too strong. It seems plausible because California was a long way from the New Jersey–based Edison company. Surely the Trust found it difficult to track and exercise patents at that remove. But the more important West Coast benefits were plentiful sunshine, and cheap labor and real estate. Many independent film producers moved there. Examples were Universal Studios, which moved soon after forming in 1912 to Hollywood, and Paramount Pictures, founded there in 1912, both still in Hollywood today. These two Yanks’ studios plus Gaumont and Pathé for the Franks are four of the five oldest surviving movie studios in the world. (The fifth is Denmark’s Nordisk Film.)

偏远的好莱坞位置并没有保护环球影城。无论如何，爱迪生信托基金紧随其后。但在 1917 年，美国最高法院以滥用专利为由裁定该信托基金不成立。最后，在 1918 年，爱迪生信托被终止为根据谢尔曼反托拉斯法非法限制贸易。67

The remote Hollywood location didn’t protect Universal Studios. The Edison Trust came after it anyway. But in 1917 the US Supreme Court found against the Trust—citing misused patents. And finally, in 1918, the Edison Trust was terminated as an illegal restraint of trade under the Sherman Antitrust Act.67

这样就完成了电影流程图诞生的扩展标题。现在我们专注于电影的那个特殊分支——动画电影——在我们的讲述中，它导致了数字大融合。通过 Digital Light 有很多途径，但我最了解的一条是通过角色动画。

This completes the extended caption for the birth of cinema flow chart. Now we concentrate on that special branch of cinema—the animated film—that leads, in our telling, to the Great Digital Convergence. There are many paths through Digital Light, but the one I know best is via character animation.

□ □ □

□ □ □ □

我们已经区分了数码灯的一半像素和制作像素的一半。到目前为止，从现实世界中提取像素来代表现实世界已经占据了我们的大部分时间。现在让像素代表虚幻世界成为我们的重点。

We have distinguished the taking pixels half of Digital Light from the making pixels half. Taking pixels from the real world to represent the real world has occupied us mostly so far. Making pixels to represent unreal worlds now becomes our focus.

一旦我们了解了计算机内部的几何模型如何成为创建虚幻世界的帧流中的像素帧，那么我们将足够了解游戏的工作原理，以及虚拟现实 (VR)、增强现实 (AR) )，以及所有应用程序和互联网的所有用户界面。我们在 3D 数字角色动画中使用的相同基本原理解释了它们。特别是，它们都共享一个正在显现的计算机内部的不可见模型的概念。在拍摄和制作之间，我们掌握了完整的数码灯。

Once we understand how a geometrical model inside a computer becomes a frame of pixels in a stream of frames that create an unreal world, then we’ll know enough to intuit how games work, and also virtual reality (VR), augmented reality (AR), and, mundanely, all user interfaces with all apps and with the internet. The same basic principles that we use in three-dimensional digital character animation explain them all. In particular, they all share the notion of an invisible model inside a computer that is being made manifest. Between taking and making, we gain a handle on full Digital Light.

现在我们开始着手制作数字光的像素部分——尤其是数字电影。让我们从电影界经典二维角色动画的简史及其工作原理开始。这为我们提供了角色动画所需的语言。然后我们在后面的章节中应用计算机强大的放大功能来达到所有的数字光。

Now we begin our runup to the making pixels part of Digital Light—digital movies in particular. Let’s start with a brief history of classic two-dimensional character animation in the film world and how it worked. This provides the language we need for character animation. Then we apply the computer’s awesome power of Amplification in later chapters to reach all of Digital Light.

动画电影：时间精神错乱

Animated Movies: Unhinged from Time

动画电影的存在时间与电影本身一样长，至少从 1895 年的发明热潮开始。但它们是什么？

Animated movies have existed as long as cinema itself, at least since that fervor of invention in 1895. But what are they?

定义它们很棘手。经典电影取材于现实世界。布景和服装可能是虚构的，灯光是人造的，演员扮演的角色可能是虚构的，但它们都是现实世界的一部分。这并不明显，但手绘动画电影的帧也来自现实世界。在卡通动画中，例如迪斯尼的《白雪公主和七个小矮人》（1937 年），每一帧都是一幅图画的照片或一幅虚幻世界的图画。那些图画或绘画是现实世界中的对象。因此，区分动画电影的不是虚幻与真实。

Defining them is tricky. Classic cinema takes its frames from the real world. The sets and costumes may be fictitious, the lighting artificial, and the characters impersonated by actors, but all of them are parts of the real world. It’s not obvious, but a hand-drawn animated movie takes its frames from the real world too. In cartoon animations, such as Disney’s Snow White and the Seven Dwarfs (1937), each frame is a photograph of a drawing or a painting of unreal worlds. Those drawings or paintings are objects in the real world. So it’s not unreal versus real that distinguishes animated movies.

区分他们的不是制造与接受。只有被称为纪录片的小类别声称拍摄现实世界本身。大部分电影都是虚构的。大多数电影的每一帧，无论是否动画，都是在用相机录制之前精心制作的。

It’s not making versus taking that distinguishes them. Only the small category known as documentary films purports to take pictures of the real world itself. Most of cinema is fictitious. Each frame of most movies, animated or not, is crafted—made—before recording with a camera.

正如漫画所暗示的那样，这也不是二维对三个维度。一些所谓的定格动画电影从真实的三维世界中获取每一帧。以雷·哈里豪森 (Ray Harryhausen) 的经典作品Jason and the Argonauts (1963) 中跳舞的骷髅为例。或者任何我喜欢和孙子们一起看的现代小羊肖恩（2007 年至今）剧集。一些最早的动画电影是定格动画，或者至少是间断动画——例如乔治·梅里爱的《闹鬼的城堡》（1896 年）或詹姆斯·斯图尔特·布莱克顿的《闹鬼的酒店》（1907 年）。一些最早的非动画电影是展示一位艺术家绘制二维卡通的电影——例如埃米尔科尔的幻想曲（1908 年）和温莎·麦凯的小尼莫（1911 年）。这些所谓的闪电素描不是动画电影——只要可以看到艺术家的身体——尽管有二维内容。

And it’s not two versus three dimensions either, as cartoons might suggest. Some so-called stop-motion animated movies take each frame from the real three-dimensional world. Take the dancing skeletons of Ray Harryhausen’s classic Jason and the Argonauts (1963), for example. Or any of the modern Shaun the Sheep (2007–present) episodes that I love to watch with my grandkids. Some of the earliest animated movies were stop-motion or, at least, interrupted motion—such as Georges Méliès’s The Haunted Castle (1896) or James Stuart Blackton’s The Haunted Hotel (1907). And some of the earliest non-animated movies were films showing an artist in the act of drawing a two-dimensional cartoon—such as Émile Cohl’s Fantasmagorie (1908) and Winsor McCay’s Little Nemo (1911). These so-called lightning sketches weren’t animated movies—so long as the artist’s body could be seen—despite the two-dimensional content.

这些电影说明动画片段或镜头经常以各种方式与非动画片段混合在一起。动画电影和他们的动画师是延伸的 Yanks vs Franks 电影奥运会的一部分。70

These films illustrate that animated snippets, or shots, were often mixed together in various ways with non-animated ones. And that animated movies and their animators were part of the extended Yanks versus Franks cinema Olympics.70

因此，区别动画电影的不是卡通与摄影现实主义。这在数字光的新世界中变得尤为明显。计算机动画帧现在可以像真实世界的照片一样具有丰富的逼真细节。

So, it’s not cartoons versus photographic realism that distinguishes animated movies. This has become particularly obvious in the new world of Digital Light. Computer-animated frames now can be as rich in photorealistic detail as photos taken of the real world.

如果这些东西都没有将动画与非动画电影区分开来，那又是什么呢？这是时间的利用。虚幻的时间，而不是虚幻的空间，定义了动画电影。动画电影与实时无关。我们称所有其他真人电影为真人电影，因为从根本上说，它们与生物一样受时间的奴役。约翰·韦恩 (John Wayne) 穿着刺耳的牛仔装大步穿过西部片场，占据了一个虚构的精心制作的空间，但他的动作被实时记录下来。

If none of these things sets animated apart from non-animated movies, then what does? It’s the use of time. Unreal time, not unreal space, defines animated movies. Animated movies are unhinged from real time. We call all others live-action movies because they are, at base, as enslaved to time as live beings are. John Wayne striding across a Western set in his jangling cowboy outfit occupies a fictitious crafted space but his movements are recorded in real time.

这可能看起来不对。众所周知，电影是导演或剪辑师及时自由排序或重新排序的一系列镜头。当然，自由地重新安排时间与被时间奴役相反。但请记住，每个镜头都是连续视觉流的时钟采样。它是现场的或实时的动作。

This might not seem right. Famously, a movie is a sequence of shots that a director or editor freely orders, or reorders, in time. Surely being free to reorder time is the opposite of being enslaved by it. But remember that each shot is a clocklike sampling of a continuous visual flow. It’s live, or real-time, action.

早期的电影只是一种奇观——嘿，看看这个，现在看看这个——一个喷嚏，一场拳击比赛，一个诱人的舞者，教皇。迪克森在 1900 年左右的许多电影都是这种类型。在电影开始之后不久，剪辑的力量的发现改变了这一切。编辑打乱了时间。我们可以以任意的时间顺序一起编辑电影片段——每个片段都有自己不可阻挡的内部时间。每个片段或镜头都是实时的奴隶，但镜头的顺序不是。有了这种自由，编辑器会导致感知时间加快或减慢。百年可以在一瞬间过去。

Early cinema was mere spectacle—hey, look at this, now look at this—a sneeze, a boxing match, a seductive dancer, the pope. Dickson’s many films circa 1900 were all of this variety. The discovery of the power of the edit, soon after those beginnings of cinema, changed all that. The edit disrupts time. We can edit together snippets of film—each with its own inexorably advancing internal time—in arbitrary temporal order. Each snippet, or shot, is a slave to real time, but the ordering of the shots isn’t. With this freedom an editor causes perceived time to speed up or slow down. Centuries can pass in an instant.

千年也是如此，如2001 年：太空漫游（1968 年），它著名地从一根骨头棍子跳到了宇宙飞船上（图 5.14）。猿猴向上投掷骨头。随着相机不断地向上和向上，它缓慢地翻滚。. . 它在空间站 V 附近变成了形状大致相同的地球卫星，这是一个雄伟的旋转中转站，供人类前往月球。导演斯坦利库布里克的“比赛剪辑”瞬间跨越了整个人类历史。

So can millennia, as in 2001: A Space Odyssey (1968), which famously jumps from a bone cudgel to a spacecraft (figure 5.14). An ape hurls the bone upward. It tumbles slowly as the camera follows it up and up and . . . it becomes an Earth satellite of roughly the same shape in the vicinity of Space Station V, a majestically rotating waystation for humans bound for the Moon. Director Stanley Kubrick’s “match cut” spans the entire history of mankind in an instant.

剪辑的力量来自于叙事和情感。由此产生的软件就是我们称之为“电影”的艺术形式。比喻说，无数可能的程序在我们的大脑上进行计算。

From the power of the edit comes narrative and emotion. And the resulting software is the artform we call “the cinema.” An infinity of possible programs compute, metaphorically, on our brains.

动画电影将剪辑发挥到了极致——每帧一个。时间不再是限制。没有视觉流程可供采样。下一帧通常是动画师打算在未来 24 秒内拍摄的当前帧，就好像它是从视觉流中实时记录的一样。但不一定是这样。它发生在动画师创建的虚构时间，可能很少或没有与实时的关系——或者是对采样实时的精确模拟。

Animated movies take the edit to the limit—one every frame. Time is no longer a restriction. There is no visual flow to sample. It’s common for the next frame to be what an animator intends to be the current frame taken one twenty-fourth of a second into the future, as if it were recorded in real time from a visual flow. But it doesn’t have to be. It occurs at a fictitious time created by the animator, that might bear little or no relationship to real time—or alternatively be a dead-on accurate simulation of sampled real time.

因此，动画电影的世界正在争夺中。它可以像艺术家希望的那样抽象。任何顺序的任何形式都是允许的。到了极限，抽象动画——纯粹来自任意视觉的情感创造，就像音乐来自任意声音一样——是等待莫扎特、肖邦和斯特拉文斯基的伟大艺术形式之一，但一直有一些实践者。著名的例子是 Walter Ruttman 的Lichtspiel: Opus I (1927) 和 Oskar Fischinger 的Motion Paint No. 1 (1947)——纯粹的视觉音乐。71

Hence the world of the animated movie is up for grabs. It can be as abstract as the artist wishes to make it. Any form in any order is permitted. Taken to the limit, abstract animation—the creation of emotion as purely from arbitrary visuals as music is from arbitrary sounds—is one of the great artforms awaiting its Mozart, Chopin, and Stravinsky, but there have been a few practitioners all along. Notable examples are Walter Ruttman’s Lichtspiel: Opus I (1927) and Oskar Fischinger’s Motion Paint No. 1 (1947)—sheer visual music.71

重温塔与臭味

Tower versus Stinks Revisited

我还没有真正解释过电影是如何运作的——尤其是动画电影。就像我们对演员和动画师如何做他们所做的事情一无所知一样，这是因为大脑几乎肯定会参与其中。在动画的一般情况和特殊情况下，这种未知领域是电影技术的极限。一旦技术通过学生，我们就不得不举手——或者做出有根据的猜测。换句话说，到依靠视觉持久性和明显运动的心理物理现象只是另一种说法，“大脑做到了”。

I haven’t really explained how films work—especially animated films. As with our ignorance of how actors and animators do what they do, it’s because the brain is almost certainly involved. That terra incognita is the limit on film technology in both the general case and the special case of animation. Once the technology passes through the pupil, we have to throw up our hands—or make educated guesses. In other words, to rely on the psychophysical phenomena of persistence of vision and apparent motion is just another way of saying, “The brain does it.”

这本书的一个主要主题是采样定理，科捷尔尼科夫的伟大想法，是理解数字光的关键。由于电影似乎以每秒 24 帧的速度对视觉流进行采样，因此采样定理肯定应该解释为什么电影有效。但我们发现它们并非完全那样工作。从抽样理论得出的可爱的理想电影系统并不是实际使用的系统。

A major theme of this book is that the Sampling Theorem, Kotelnikov’s great idea, is key to understanding Digital Light. Since films appear to sample visual flow at 24 frames per second, surely the Sampling Theorem should explain why films work. But we discovered instead that they don’t work exactly that way. The lovely ideal movie system that follows from sampling theory was not the one that was actually used.

实际的系统绝对是某种采样过程，但它每帧投影两次，而且帧很胖，充满了诸如运动模糊之类的线索，供大脑享用。在理想的系统中，框架很薄——瞬时的。因此，它们不能包含运动模糊线索。但它们可以表示任何不超过两倍最高频率限制的运动。采样定理保证了这一点。当我们接近 Digital Light 时，我们将寻找对理想系统的遵守。现代现实世界是否在瞳孔之外传播和添加样本？还是继续依靠“大脑做它”的手波？

The actual system was definitely a sampling process of some sort, but it projected each frame twice, and the frames were fat and filled with clues such as motion blur for the brain to feast on. In an ideal system the frames are thin—instantaneous. They therefore cannot contain motion blur clues. But they can represent any motion that doesn’t exceed the twice-the-highest-frequency restriction. The Sampling Theorem guarantees it. As we approach Digital Light, we’ll look for adherence to the ideal system. Does the modern real world spread and add the samples outside the pupil? Or continue to rely on “the brain does it” handwave?

动画电影甚至超出了采样定理的范围——尤其是那些用 2 秒拍摄的。在这种情况下，每一帧都被投射到视网膜上四次！考虑弹跳球。它与地面接触的那一刻一次又一次地发生。然而，我们的大脑会弄清楚。它没有看到四重打击，而是感受到了一种持久的美味冲击——毫无疑问，这得益于动画大师的技巧，如挤压和拉伸、预期和夸张。

Animated films fall even farther outside the bounds of the Sampling Theorem—especially those shot on 2s. In that case, every frame is projected onto the retina four times! Consider the bouncing ball. The moment of its contact with the ground happens again and again and again and again. Yet our brain figures it out. It doesn’t see the quadruple whammy, but rather perceives one long delicious impact—aided undoubtedly by the tricks of the animation masters, like squash and stretch, anticipation and exaggeration.

理想与现实暗示了塔与臭，这是本书的另一个主题。它们从根本上都很重要。事实上，这正是塔与臭的对比，以及为什么将一个人的创造力评价为另一个人的创造力是错误的。真正的电影机器是在臭气熏天的环境中创造出来的。我们可以稍后在象牙塔中出现，并提出一个美丽的数学想法来“解释”电影机器工作的原因。但很明显，电影发明者并不是从理论开始的。采样定理不能解释他们的机器。但它确实告诉我们下一步该走哪条路。这种行为和思考的相互作用比理论家或工程师通常愿意承认的更接近世界的运作方式。

Ideal versus actual is suggestive of tower versus stinks, another theme of this book. They both fundamentally matter. Indeed, it’s exactly the tower versus the stinks, and why it’s wrong to rate the creativity of one over that of the other. Actual movie machines were created in the stinks. We can come along later in the ivory tower and propose a beautiful mathematical idea to “explain” why movie machines work. But it’s clear that the movie inventors didn’t start from theory. The Sampling Theorem doesn’t work as an explanation of their machines. But it does show us which way to go next. This interplay of doing and thinking is closer to the way the world works than either theoreticians or engineers commonly like to admit.

电影和动画的故事也生动地让我们想起了电影技术的另一部分秘密，一个我们更不了解的秘密。沙利文和迪斯尼业务的关键在于他们聘请了具有稀有艺术天赋的人来使照片栩栩如生。这也是在数字世界中有效的方法。我一生中最好的雇佣是迪斯尼训练有素的动画师约翰拉塞特，他莫名其妙的天赋是皮克斯无与伦比的早期成功的无可置疑的关键。

The story of film and animation also vividly reminds us of another part of the secret of movie technology, one we understand even less. The key to both the Sullivan and Disney businesses was the fact that they hired people with the rare artistic talent to make pictures come alive. That’s the method that works in the digital world too. The best hire of my life was the Disney-trained animator John Lasseter, whose inexplicable talent was the unquestioned key to Pixar’s unmatched early success.

摩尔定律因子现在（2020 年）为 1000 亿X，并迅速接近 1 万亿X。尽管所有关于人工智能和深度学习的主张都受到了计算能力的超新星爆发的启发，但我们现在几乎没有比 1920 年代和 1930 年代更接近于理解动画师或演员是如何创造个性的。根据电视广告，现在所有的汽车都是智能的，但没有一辆汽车能激发一个人物的灵感。

The Moore’s Law factor is now (2020) 100 billion X and rapidly approaching 1 trillion X. Despite all the claims about artificial intelligence and deep learning inspired by that awesome, supernova surge in computational power, we are hardly any closer now to understanding how an animator or actor creates a personality than we were in the 1920s and 1930s. According to television commercials, all automobiles are now intelligent, but none of them can inspirit a character.

我并不是说技术存在硬性限制——机器不能具有创造力。我个人的信念是，我们总有一天会了解创造者是如何创造甚至是机器的。但那是信仰，而不是科学。如果有足够的理由，我可以选择相信相反的观点——许多人也这样做。声称或暗示我们几乎就在那里，现在任何一天都是“基于信仰的科学”。与此同时，在可预见的未来，我们将继续聘请最优秀的人类艺术人才来制作动画和表演——他们是唯一已知的解决方案。我们仍然在寻找世界上为数不多的 Ubs、Ottos 和 Lasseters。

I am not saying that there exists a hard limit to technology—that machines cannot be creative. My personal faith is that we will someday understand how creators create and perhaps even machines. But that’s faith, not science. I could, with as much justification, elect to believe just the opposite—and many humans do. It’s “faith-based science” to claim or imply that we are almost there, any day now. Meanwhile, into the foreseeable future, we continue to hire the best human artistic talent we can to animate and act—they’re the only known solution. We still seek out the few Ubs, Ottos, and Lasseters of the world.

样条形状

The Spline Shape

旧金山地区的一大乐趣是轻松享受美妙的音乐。即便如此，当伟大的印度西塔琴演奏家 Ravi Shankar 在 1982 年的一个凌晨时分走进卢卡斯影业的计算机图形实验室时，我还是感到震惊。我独自一人在我们的一位常驻天才编写的绘画程序上工作，汤姆波特。那天晚上我一直致力于了解 Tom 的最新功能——样条曲线。Ravi 的主持人是 Clark Higgins，他是 Grateful Dead 摇滚乐队和我们的视频向导。2

A pleasure of the San Francisco area is easy access to fine music. Even so, I was astounded when Ravi Shankar, the great Indian sitar player, walked into the computer graphics lab at Lucasfilm in the wee hours of a night in 1982. I was working alone on the paint program written by one of our resident geniuses, Tom Porter. I had devoted that evening to learning about Tom’s latest feature—splines. Ravi’s host was Clark Higgins, the video wizard for both the Grateful Dead rock band and us.2

“你认识拉维·香卡，不是吗，阿尔维？”

“You know Ravi Shankar, don’t you, Alvy?”

我融化了。“我当然是了。” 我第一次见到他是在 60 年代的一部决定性电影《蒙特雷流行》（1968 年）中。我知道他曾于 1969 年在伍德斯托克演出，这是那个时代的另一个决定性事件。我不认识他，但我肯定认识他。

I melted. “Of course I do.” I had seen him first in Monterey Pop (1968), a defining movie of the Sixties. I knew he’d performed at Woodstock in 1969, another defining event of the era. I didn’t know him, but I certainly knew him.

拉维是一个漂亮的男人，身材娇小，面带微笑，穿着（大部分）白色印度服装无可挑剔。最重要的是，他身上散发着一种微妙而令人惊讶的香味——一种感性的光环，既增强了他的存在感，又在他站在我的肩膀上凝视时将我吞没。

Ravi was a beautiful man, small, smiling, and impeccably dressed in (mostly) white Indian apparel. Best of all, he wore a subtle and surprising fragrance that surrounded him—a sensual aura that both enhanced his presence and engulfed me as he stood peering over my shoulder.

碰巧在靠近顶部的电脑屏幕上显示了一朵花。前一天一位来访的艺术家在那里画了一朵红色和蓝色花瓣的小花，中心有一个白点。当拉维加入我时，它照亮了房间，就像一朵真正的花一样。

By happenstance there was a flower displayed on the computer screen, near the top. A visiting artist the previous day had painted a small blossom there of red and blue petals and a white-dotted center. It brightened the room when Ravi joined me, like a cutting of a real flower.

“样条曲线，”我解释说，“是通过几个点的优美曲线。我们的计算机只能记录这条路径上的几个点。” 我做了一个盛大的手势，用手写笔在一个大平板电脑上划过。那天晚上我没说什么。我只是向拉维展示了它是如何工作的。

“A spline,” I explained, “is a graceful curve through a few points. Our computer can register only a few points along this path.” With a grand gesture, I stroked a stylus across a large tablet. I said little else that night. I simply showed Ravi how it worked.

手势是任意的，从低处开始向上移动。手腕一甩，我在中风时短暂地向下扫过，然后再次向上扫过，最终结束。

The gesture was an arbitrary one, starting low and moving upward. With a wrist flick, I swept briefly downward in midstroke, then upward again, ending high.

然后汤姆的绘画程序立即接管了。首先，它通过平板电脑的稀疏点创建了一条样条曲线。请注意，我们看不到样条曲线。它是纯几何图形，宽度为零，存储在计算机的某个地方。但随后，在稍作停顿之后，程序将其渲染为我们面前彩色显示屏上的一笔画。不可见的样条线成为可见笔画的支柱。我的手势和程序的笔划之间的延迟是如此短暂，以至于似乎——带着一点想象力——我，而不是电脑，直接在屏幕上画了它。颜色是随机选择的——棕色，幸运的是。

Then Tom’s paint program instantly took over. First, it created a spline through the sparse points from the tablet. We couldn’t see the spline, mind you. It was pure geometry, with zero width, stored inside the computer somewhere. But then, after a hint of a pause, the program rendered it into what looked like a stroke of paint on the color display before us. The invisible spline became the backbone of a visible paint stroke. The lag between my gesture and the program’s stroke was so brief that it seemed—with a pinch of imagination—that I, not the computer, had painted it directly on the screen. The color had been a random choice—a brown, as luck would have it.

该程序模拟了具有柔软边缘和不同宽度的笔触。笔画在显示屏底部附近开始时没有宽度，然后随着向上扫过并进入曲线逐渐变宽，最后在向上结束时再次变窄。. . 只是在花！一个完美的茎。好像我是这样计划的。但我没有。我们三个人都知道我没有。它让我们大吃一惊。在那个漫长的时刻，我的音乐英雄拉维·香卡（Ravi Shankar）和我一起沉浸在他自己的天堂般的气味中，和我一样震惊，他摸了摸我的肩膀，低声说：“Allllllllvyyyyy！”

The program simulated a brushstroke with soft edges and varying width. The stroke began with no width near the bottom of the display, then widened gradually as it swept upward and into the curves, and finally narrowed to nothing again as it ended upward . . . just at the flower! A perfect stem for it. As if I had planned it that way. But I hadn’t. All three of us knew I hadn’t. It took our breaths away. In that extended moment Ravi Shankar, my musical hero, swimming with me in his own heavenly scent and as startled as I, touched my shoulder and whispered, “Alllllllvyyyyy!”

一群鸭子，一群鲸鱼

A Plump of Ducks, a Pod of Whales

Ravi Shankar 演示的样条曲线是源自同名绘图员工具的几何图形。早在个人电脑出现之前，我父亲就教我机械绘图。这些工具是用于画圆的精密德国罗盘、用于直角和直线的 T 形方尺以及印度墨水笔和笔尖。他有另一个相当神秘的工具，叫做法式曲线（图 6.1），如果你把图片倒过来，它就类似于圣诞老人的雪橇。它的目的是在我们试图绘制一条不寻常的平滑曲线时稳定我们颤抖的手。在插图中，绘图员将法式曲线放置在三个连续的位置，以帮助他分段绘制曲线。这些部分是曲线部分 A、B 和 C，每个部分都位于法国曲线的许多边缘的不同位置。这位绘图员很幸运。他找到了所有需要的曲线沿着圣诞老人的雪橇滑道，彼此靠近。但有时你找不到你所寻找的曲线。

The spline of the Ravi Shankar demo is a piece of geometry that derives from a draftsman’s tool of the same name. My father taught me mechanical drawing as a child, long before the personal computer. The tools were a precision German compass for drawing circles, a T-square for right angles and straight lines, and India ink pens and nibs. He had another rather mysterious tool called a French curve (figure 6.1) that resembled Santa’s sleigh if you turn the picture upside down. Its purpose was to steady our shaky hands as we tried to draw a smooth curve of unusual shape. In the illustration a draftsman has placed a French curve in three successive positions to help him draw a curve piecewise. The pieces are curve parts A, B, and C, each found in various places along the French curve’s many edges. This draftsman was lucky. He found all needed curve parts near one another along the runner of Santa’s sleigh. But sometimes you can’t find the curve you seek.

认真的绘图员需要更值得信赖的东西。豪华游艇的设计师不会忍受可能缺少所需曲线部件的法国曲线。例如，为了绘制赛车船体的曲线，他们使用一个用重物固定在适当位置的薄柔性条带，如图 6.2（右）所示。条带通过重物固定的点并通过它们松弛成平滑的曲线，因此条带的边缘可以追踪到船计划中。那个绘图工具——条带和砝码——被称为样条线。4

Serious draftsmen need something more trustworthy. Designers of luxury yachts won’t put up with a French curve that might lack the curve parts needed. To draft curves for a racing hull, say, they use a thin flexible strip held in place with heavy weights, as shown in figure 6.2 (right). The strip passes through points fixed by the weights and relaxes into a smooth curve through them so the edge of the strip can be traced into the boat plans. That drafting tool—the strip and the weights—is called a spline.4

Ravi Shankar 演示中的几何样条直接受到船设计师的物理启发。出于显而易见的原因，物理版本中的重量被称为鸭子（或鲸鱼）。如图所示，您甚至可以购买画成绿头鸭、秋沙鸭、丑角等的丰满鸭子。从物理鸭子突出的钩尖是物理样条必须通过的点。几何样条只有虚拟鸭子，没有实体或钩子。在这种情况下，鸭子只是样条必须通过的点。在 Ravi Shankar 演示中，鸭子是我做出手势时从平板电脑获得的分离点。

The geometrical spline in the Ravi Shankar demo was directly inspired by the boat designer’s physical one. The weights in the physical version are known as ducks (or whales), for obvious reasons. You can even buy a plump of ducks painted as mallards, mergansers, harlequins, and so forth, as shown. The hook tips protruding from physical ducks are the points that the physical spline must pass through. A geometrical spline has only virtual ducks, without bodies or hooks. The ducks in this case are simply the points that the spline must pass through. In the Ravi Shankar demo, the ducks were the separated points obtained from the tablet as I made my gesture.

计算机图形定义

Computer Graphics Defined

Ravi Shankar 样条演示是计算机图形学的一个示例，它是 Digital Light 的一个主要分支——一个合成分支，可以制作图片而不是拍摄它们。计算机图形学总是需要两个步骤：我们通过在 Creative Space 中使用不可见的几何图形建模来创建对象，这一过程我将在本章中详细描述下一个。然后我们通过将基于几何的模型渲染到显示空间的散布像素中来查看它们，例如打印页面、手机显示器或虚拟现实 (VR) 护目镜。这就是我们对计算机图形的定义。

The Ravi Shankar spline demonstration is an example of computer graphics, a major branch of Digital Light—a synthetic branch, making pictures not taking them. Computer graphics always requires two steps: We create objects by modeling them with invisible geometry in Creative Space, a process I’ll describe at length in this chapter and the next. Then we view them by rendering the geometry-based models into the spread pixels of a Display Space, such as a printed page, a cellphone display, or virtual reality (VR) goggles. That’s our definition of computer graphics.

在与 Ravi Shankar 合作的那个晚上，Tom Porter 的程序创建了一个不可见的样条线作为笔画的主干。那是笔画背后的几何形状。尽管几何是计算机图形定义的基础，但模型不仅仅是几何。笔画模型还具有颜色、柔和的半透明边缘和随时间变化的宽度。该程序将笔画模型渲染为像素，然后通过卢卡斯影业图形实验室的全彩显示器传播到可见性。这是二维计算机图形的一个例子，因为该模型是基于二维几何的。

On that night with Ravi Shankar, Tom Porter’s program created an invisible spline as the backbone of a paint stroke. That was the geometry underlying the stroke. Although geometry is fundamental to the definition of computer graphics, a model isn’t only geometry. The stroke model also had color, soft semi-transparent edges, and a width that varied in time. The program rendered the model of the stroke into pixels which were then spread into visibility by the full-color display in the Lucasfilm graphics lab. It was an example of two-dimensional computer graphics, because the model was based on two-dimensional geometry.

本章中同一样条曲线的图片是一个更简单的渲染。它的模型与我向 Ravi Shankar 展示的笔画具有完全相同的几何路径。但是该模型的其他组件是不同的。它是白色背景上等宽的黑线。两张图片都是计算机图形的示例。创意空间中的单个几何对象可以在显示空间中以多种方式呈现，具体取决于那些非几何模型组件。10

The picture in this chapter of that same spline is a much simpler rendering. Its model has exactly the same geometrical path as the paint stroke I showed to Ravi Shankar. But the other components of the model are different. It’s a black line of constant width on a white background. Both pictures are examples of computer graphics. A single geometric object in Creative Space can be rendered a great many ways in Display Space, depending on those non-geometrical model components.10

另一方面，如果没有几何图形，那么它就不是计算机图形学。Adobe Illustrator 将二维几何模型渲染为像素。这是一个计算机图形程序。Adobe Photoshop 从头开始​​创建像素或修改从现实世界中获取的像素。它不是计算机图形程序。这是一个图像处理程序，来自 Digital Light 的另一个主要分支。同样，早期的 Apple 提供了 MacDraw（一种计算机图形程序）和 MacPaint（一种像素打包或图像处理程序）。（有关“几何”的广义概念的讨论，请参见注释。）

On the other hand, if there’s no geometry, then it’s not computer graphics. Adobe Illustrator renders two-dimensional geometric models into pixels. It’s a computer graphics program. Adobe Photoshop creates pixels from scratch or modifies pixels taken from the real world. It’s not a computer graphics program. It’s an image processing program, from another major branch of Digital Light. Similarly, early Apple offered MacDraw, a computer graphics program, and MacPaint, a pixel-packing, or image processing, program. (See the annotation for discussion of a generalized notion of “geometry.”)

计算机图形是几何输入，像素输出。图像处理是像素输入，像素输出。两者都用于创建图片。在内存非常昂贵的早期，图像处理只是为了拍照，但摩尔定律使得在新千年的现在，直接从像素制作照片成为可能。

Computer graphics is geometry in, pixels out. Image processing is pixels in, pixels out. Both are used for creating pictures. In the early days of very expensive memory, image processing was about taking pictures only, but Moore’s Law has made it possible now, in the new millennium, to make pictures directly from pixels.

Adobe 和 Apple 产品对反映了我在第 4 章中提到的书法“绕道”造成的旧数字光划分。早期，基于几何的程序设计用于书法显示，而基于像素的程序用于光栅显示。现在使区分变得棘手的是，自从大数字融合以来，这两个品种都以像素显示。计算机图形与图像处理的区别在于其独特的基于几何的创意空间的内容。但是两个分支现在共享相同的显示空间。

The Adobe and Apple product pairs reflect the old division of Digital Light caused by the calligraphic “detour” I mentioned in chapter 4. In the early days, geometry-based programs were designed for calligraphic displays, and pixel-based programs for raster displays. What makes the distinction tricky now is that, since the Great Digital Convergence, both varieties display in pixels. What separates computer graphics from image processing is the contents of its distinctive geometry-based Creative Space. But both branches now share the same Display Space.

有很多二维计算机图形的例子。甚至我现在使用的 Microsoft Windows 界面也是计算机图形。窗户开着显示空间中的屏幕是程序创意空间中桌面上一堆文件的矩形模型。与 Microsoft Word 的界面类似。它的模型是代表文档页面的一系列矩形，页面上的每个字母最终都由一个几何模型定义。网页设计和使用结果页面的浏览器是进一步的例子。几何输入，像素输出。

There are many examples of two-dimensional computer graphics. Even the interface to Microsoft Windows that I’m using right now is computer graphics. The windows on the screen in Display Space are rectangular models of a stack of papers on a desktop in the program’s Creative Space. And the interface to Microsoft Word is similar. Its model is a sequence of rectangles representing the document pages, and each letter on the pages is defined ultimately by a geometric model. Webpage design and the browsers that use the resulting pages are further examples. Geometry in, pixels out.

本书的一个主要目标是展示一部类似皮克斯的电影是如何从计算机中出现的。这种电影的每一帧都用计算机内部的三维几何建模，然后渲染成二维扩展像素以进行全彩显示。这就是我们所说的三维计算机图形学，这本书的主要焦点。电子游戏的创建方式相同。VR 也是如此——每帧有两个显示器，每只眼睛一个，以产生立体效果。

A major goal of this book is to show how a Pixar-like movie emerges from a computer. Each frame of such a movie is modeled with three-dimensional geometry internal to the computer, then rendered into two-dimensional spread pixels for full-color display. That’s what we mean by three-dimensional computer graphics, the main focus of this book. Videogames are created the same way. And VR too—with two displays of each frame, one for each eye, to produce a stereo effect.

样条曲线是曲线，而不是曲面，我们现在要将注意力集中在曲面建模上。我们考虑的下一个形状用于此目的。其中第一个简单而熟悉。

A spline is a curve, not a surface, and we want to focus attention now on modeling surfaces. The next shapes we consider serve that purpose. The first of them is simple and familiar.

三角形

The Triangle Shape

究竟什么是三角形？我查阅了我在新墨西哥州克洛维斯高中时精心保存的平面几何教科书，这本书塑造了我的整个未来。本书从第 7 页开始，以点和直线开头。这两个概念无法真正定义——我们只是“知道”它们是什么——但我们可以通过它们缺少什么或不具备什么来开始“理解”它们：一个点没有宽度、没有高度和没有厚度。这是纯粹的位置。一条直线没有宽度，没有厚度，并且在一维的两个方向上都趋于无穷大。可以通过命名其上的任意两个点来指定。线段是由直线上的两个点（线段的端点）定义的有限段直线。折线由线段组成，除了最后一条线段外，每条线段都以头尾相连。11

What exactly is a triangle? I consulted the carefully preserved plane geometry textbook from my Clovis, New Mexico, high school, a book that shaped my entire future. The book begins on page 7 with points and straight lines. The two ideas can’t really be defined—we just “know” what they are—but we can start to “understand” them by what they lack, or what they are not: A point has no width, no height, and no thickness. It’s pure location. A straight line has no width, no thickness, and goes to infinity in both directions of its one dimension. It can be specified by naming any two points on it. A line segment is a finite piece of a straight line defined by two points on the line, the line segment’s endpoints. A broken line is composed of line segments, each connected nose-to-tail with one other, except the last one. So a broken line has two dangling endpoints.11

令我惊讶的是，教科书直到第 57 页才到达低三角形。它说，多边形是一条闭合的折线。最后一条线段的最后一个端点连接到第一条线段的第一个端点——没有悬垂端点。最后，三角形是只有三个边的多边形。这些都不是可见的，但教科书确实说一个点由图片中的一个点表示。图 6.8 在一张图中显示了几何课。这些是我刚才提到的五个不可见几何对象的可视化。它们是简单抽象几何模型的具体渲染。

To my surprise, the textbook doesn’t arrive at the lowly triangle until page 57. A polygon is a closed broken line, it says. The last endpoint of the last line segment is connected to the first endpoint of the first line segment—no dangling endpoints. Finally, a triangle is a polygon with just three sides. None of these is visible, but the textbook does say that a point is represented by a dot in a picture. Figure 6.8 shows that geometry lesson in one figure. These are visualizations of the five invisible geometrical objects I just mentioned. They are concrete renderings of the simple abstract geometric models.

教科书对三角形的冗长方法确实具有引入一个重要术语的优点。它是多边形，计算机图形学中一个非常受欢迎的词。计算机图形学的一般方法是用多边形描述一个虚构的世界。

The textbook’s longwinded approach to the triangle does have the advantage that it introduces an important term. It’s polygon, a much beloved word in computer graphics. A general approach in computer graphics is to describe a fictitious world with polygons.

但是我们将立即摆脱对多边形一词的需求，因为这个关键事实：任何多边形都可以细分为三角形：选择多边形的任何角或顶点，并用线将其连接到所有其他角（顶点）段。所以，我们只需要谈论三角形。图 6.9 显示了将四边形多边形转换为两个三角形的两种不同方法，并带有一条额外的线段（虚线）。五边形多边形需要两条额外的线，依此类推。（注意事项见注释。）12

But we’ll immediately rid ourselves of the need for the word polygon with this key fact: Any polygon can be subdivided into triangles: choose any corner, or vertex, of a polygon and connect it to all the other corners (vertices) with line segments. So, we only ever have to talk about triangles. Figure 6.9 shows two different ways to convert a four-sided polygon into two triangles with one additional line segment (dotted). It would take two additional lines for a five-sided polygon, and so forth. (See the annotation for caveats.)12

由于这个技巧，我们只需要专注于由三角形组成的模型。不过，并不是每个模型都可以用三角形制作。Ravi Shankar 曲线的几何模型只是样条曲线，而不是多边形。第一张数码图片“First Light”中的文字，以及“数码之光的黎明”一章中早期电子游戏的游戏板也有简单的模型，并未简化为三角形。除了少数例外，这些模型只是直线段。

Because of this trick, we only need to concentrate on models made of triangles. Not every model can be made with triangles, though. The geometric model for the Ravi Shankar curve is just a spline, not a polygon. The text in the first digital picture, First Light, and the game boards of the early videogames in the Dawn of Digital Light chapter also had simple models that didn’t reduce to triangles. The models were, with few exceptions, just straight line segments.

但是考虑一下皮克斯的动画电影，从玩具总动员到超人总动员 2、Illumination 的卑鄙的我，以及所有其他具有 3D 角色的动画电影——“皮克斯式”电影。它们的特征是由曲面组成的，曲面可以只用三角形来建模。

But consider Pixar’s animated films from Toy Story to Incredibles 2, Illumination’s Despicable Me, and all other animated films with three-dimensional characters—“Pixar-like” movies. Their characters are composed of surfaces, and surfaces can be modeled just with triangles.

如果我们学会了如何处理一个三角形，那么我们就可以让 Amplification（计算机的伟大荣耀）来处理所有其余部分，通常是数百万个。本章（和下一章）的中心思想是，如果你能理解一个三角形是如何从计算机内存到显示屏的，那么通过调用 Amplification您可以直观地了解完整图片是如何进入屏幕的。正是计算机不知疲倦地一遍又一遍地计算同一件事，数百万甚至数十亿次，速度非常快，这使得计算机图形成为可能。这很难做到——对于一个没有帮助的人来说是不可能的——但很容易理解。

If we learn how to deal with a single triangle, then we can let Amplification—that great glory of computers—take care of all the rest, often many millions of them. The central idea of this chapter (and the next) is that if you can understand how one triangle gets from computer memory to display screen, then by invoking Amplification you can intuit how a full picture gets to the screen. It’s the computer tirelessly computing the same thing over and over, millions even billions of times, very fast, that makes computer graphics possible. It’s hard to do—impossible for an unaided human—but easy to understand.

早期计算机图形学流程图（续）

Flow Chart of Early Computer Graphics (Cont.)

现在让我们将这些发展置于历史背景中。我在这里使用了一个流程图，就像我在前两章中使用的那样，来引导我们了解涉及许多参与者的复杂历史。我在“数字光的黎明”一章中介绍的计算机历史流程图涵盖了 1940 年代末和 1950 年代初。我们可以将那个时期视为摩尔定律之前时期的第一阶段——即第一纪元的第一阶段。图 6.13 是该图表（图 4.6）到 1950 年代后期和 1960 年代大部分时间的延续， Epoch 1 的第二阶段。此处重复标有“Whirlwind”的框以显示两个图表的连接位置。

Let’s now turn to placing these developments in historical context. I use a flow chart here, like those I used in the preceding two chapters, to guide us through a complex history involving many players. The flow chart of the history of computers I presented in the Dawn of Digital Light chapter covered the late 1940s and the early 1950s. We can think of that period as the first stage of the period before Moore’s Law—that is, the first stage of Epoch 1. Figure 6.13 is the continuation of that chart (figure 4.6) into the late 1950s and through much of the 1960s, the second stage of Epoch 1. The box labeled “Whirlwind” is repeated here to show where the two charts join.

“Moore's Law 1965” 项出现在流程图的中下部，表示启动 Epoch 2 的重大事件。我们将在下一章中保留对 Epoch 2 的探索，除了为了方便起见，此处包含一个经过仔细标记的事件. （发生在 1965 年之后的事件并不意味着它利用了摩尔定律。）个人被包围在圆圈中，而群体被包围在椭圆中。计算机和专用硬件设备位于矩形中。程序、书籍和概念——所有软想法——都是平行四边形。

The item “Moore’s Law 1965” appears in the lower middle of the flow chart, indicating the fateful event that kicked off Epoch 2. We’ll save exploration of Epoch 2 for the next chapter, except for one carefully labeled event included here for convenience. (That an event occurred after 1965 doesn’t imply that it utilized Moore’s Law.) Individuals are enclosed in circles, and a group in an ellipse. Computers and a special-purpose hardware device are in rectangles. Programs, books, and concepts—all soft ideas—are in parallelograms.

我在前几章中所说的关于流程图的所有内容也仍然适用于这一章：这不是对该领域的详尽介绍。此处省略的许多球员和事件在文本的其他地方或注释中提供 - 但肯定不是全部。本章的散文可以被认为是图表的冗长标题。和以前一样，图表的复杂性表明，高科技的故事很少，如果有的话，是一个单一领导者的简单叙述。17

Everything I said about flow charts in earlier chapters continues to hold for this one too: It’s not an exhaustive presentation of the field. Many players and events that are omitted here are supplied elsewhere in the text or in the annotations—but certainly not all of them. And the chapter’s prose can be thought of as a lengthy caption for the chart. As before, the chart’s complexity demonstrates that the story of a high technology is seldom, if ever, a simple narrative of a single leader.17

一个重要的区别：物体与图片

An Important Distinction: Objects versus Pictures

根据定义，计算机图形是从计算机内存中保存的不可见几何模型创建可见图片。使用这些模型有两种主要方法，这种区别导致计算机图形的主要划分。

Computer graphics by definition is the creation of visible pictures from invisible geometric models held in computer memory. There are two major ways to use those models, and that distinction leads to a major partitioning of computer graphics.

在一种情况下，模型表示旨在存在于现实世界中的对象。计算机辅助设计(CAD) 从业者使用计算机图形来设计对象。归根结底，对象在 CAD 中很重要，而不是图片。CAD是面向对象的计算机图形学。

In one case, the models represent objects intended to exist in the real world. A computer-aided design (CAD) practitioner uses computer graphics to design objects. Ultimately, objects matter in CAD—not pictures of them. CAD is object-oriented computer graphics.

相反的情况是面向图片的计算机图形学。数字电影是一个很好的例子，说明只有图片才是重要的——而不是所描绘的物体。

The opposite case is picture-oriented computer graphics. A digital movie is an excellent example of how only pictures matter—not the objects pictured.

一个主要的附带条件是将 CAD 与面向图片的计算机图形学分开。一句话，就是准确。CAD 中对象的计算机模型是旨在存在于现实世界中的对象的准确表示，例如汽车、建筑物和机器零件。早期，CAD 生成的是对象，而不是图片。例如，计算机模型可用于驱动铣床，从金属、泡沫或木块上切割出相应的物体。

A major proviso separates CAD from picture-oriented computer graphics. In a word, it’s accuracy. The computer models of objects in CAD are accurate representations of objects intended to exist in the real world, such as cars, buildings, and machine parts. In the early days, CAD produced objects, not pictures. For example, a computer model might be used to drive a milling machine that cut the corresponding object from blocks of metal, or foam, or wood.

由于计算机辅助设计对象必须承受现实世界的磨损，因此它们通常在仍处于计算机模型形式时接受严格的测试。因此，在锻造和组装实际的桁架之前，可能首先在应力的计算机模拟中测试由钢桁架制成的桥梁。桥梁的精确计算机模型用作模拟交通或风对桥梁的预期载荷的输入。或者在计算机中检查打算刚性连接的航天器的两个部分模型形式，以确保它们在金属中实际实现时能够可靠地组合在一起。

Since computer-aided design objects must sustain real-world wear and tear, they are often subjected to serious testing while still in computer model form. So a bridge made of steel trusses might first be tested, in a computer simulation of stresses, before the actual trusses are forged and assembled. The accurate computer model of the bridge serves as input to the simulation of expected loading of the bridge by traffic or wind. Or two parts of a spacecraft intended to attach rigidly are checked, while still in computer model form, to ensure that they will reliably fit together when actually realized in metals.

但在其他计算机图形学中，图片即输出就是一切——因此是面向图片的计算机图形学。准确的测量和测试是无关紧要的。图片只需要看起来令人信服。皮克斯可能会将伍迪的《玩具总动员》模型发送给某人，后者将其转化为准确的、可构建的 CAD 模型并制作玩具，但原始模型的主要目的是电影《玩具总动员》。

But in the rest of computer graphics, pictures-as-output are everything—hence picture-oriented computer graphics. Accurate measurements and testing are irrelevant. The pictures only have to look convincing. Pixar might send Woody’s Toy Story model to someone who turns it into an accurate, buildable CAD model and makes a toy, but the principal purpose of the original model was Toy Story the movie.

好莱坞以其虚假的战线而闻名。想想牛仔射击游戏中的主要街道。计算机生成的电影都是虚假的前线。伍迪的表面里面什么都没有。他只是肤浅。里面没有什么可测试或压力的。谁在乎他是否可以建造？字符只需要看起来正确。

Hollywood is famous for its false fronts. Think of the main street in a cowboy shoot-’em-up. Computer-generated movies are all false fronts. There’s nothing inside Woody’s surfaces. He’s only skin deep. There’s nothing to test or stress inside. Who cares if he’s buildable? Characters only have to look right.

C.R.T. STORE考虑一下第一光的文本和HELLO MR. MURROW数字光的黎明一章中呈现的早期动画。现实世界中没有任何对象与这些显示相对应。或者回想一下同一章节中的两个早期游戏，跳棋和井字游戏。同样，现实世界中不存在与这些相对应的实际游戏板或棋子。Whirlwind 上显示的数学曲线并不是现实世界中的物体。除了在计算机显示器上散布像素和它们的（简单）不可见模型之外，这些早期示例从未有任何形式。因此，来自数字光黎明的所有图片都是面向图片的计算机图形实例，而不是计算机辅助设计。然而我们会发现，计算机图形学的最早领导者通常来自 CAD。

Consider the texts C.R.T. STORE of First Light and HELLO MR. MURROW of the early animation presented in the Dawn of Digital Light chapter. No objects in the real world corresponded to those displays. Or recall the two early games from that same chapter, checkers and tic-tac-toe. Again, no actual game board or pieces corresponding to these ever existed in the real world. And the mathematical curves displayed on Whirlwind weren’t objects in the real world. There was never any form to these early examples other than spread pixels on a computer display and (simple) nonvisible models of them. Thus all the pictures from the Dawn of Digital Light were instances of picture-oriented computer graphics, not computer-aided design. Yet we’ll find that the earliest leaders of computer graphics generally came from CAD.

计算机辅助设计和面向图片的计算机图形学密切相关，并且有着相互交织的历史。他们经常混为一谈也就不足为奇了。在本书中，计算机图形学通常意味着“图片为王”，并缩写为面向图片的计算机图形学。我小心地将“物为王”模式指定为计算机辅助设计。需要重复一遍：虽然 CAD 是计算机图形学的一部分，但我在本书中通常指的计算机图形学是非 CAD 部分。

Computer-aided design and picture-oriented computer graphics are closely related and have intertwined histories. It’s no surprise that they are often conflated. From here on in this book, computer graphics will usually mean “pictures are king” and abbreviate picture-oriented computer graphics. I carefully designate the “objects are king” mode as computer-aided design. That bears repeating: although CAD is part of computer graphics, I’ll usually mean by computer graphics in this book the non-CAD part.

我限制了前几章中的图表来绘制通向数字光的路径。该图表（图 6.13）仅进一步关注那些导致 Digital Light 的计算机图形部分的内容。它强调了对制作数字电影特别重要的计算机图形部分，这是下一章的重点。幸运的是，它仍然捕捉到了早期计算机图形学的许多重大事件。

I constrained the charts in previous chapters to map the pathways leading to Digital Light. This chart (figure 6.13) concentrates further only on those that led to the computer graphics part of Digital Light. It emphasizes those parts of computer graphics that are especially important for making digital movies, the focus of the next chapter. Fortunately, it nevertheless captures many major events of early computer graphics.

环境

The Milieu

计算机图形学出现在特定环境中。它与个人计算机和互联网的发明共享。这是一个国家技术蓬勃发展的故事。

Computer graphics emerged in a particular milieu. It shares it with the inventions of the personal computer and the internet. It’s quite a story—of a national technological blossoming.

数字恐龙时代

The Age of Digital Dinosaurs

数字光在 1945 年美国原子弹和 1949 年苏联氢弹之间——二战和冷战之间——出现了。回想一下“数码光的黎明”一章，在 1940 年代在少数珍贵的计算机上拍照被认为是轻浮的。他们“应该”计算原子弹或氢弹的统计数据。尽管如此，1947 年，第一张数码照片 First Light 出现在 Manchester's Baby。1951 年和 1952 年在曼彻斯特和剑桥大学创建了两个最早记录的交互式视频游戏。1951 年，由空军资助的第一个有记录的计算机动画出现在麻省理工学院的旋风上。在所有这些早期案例中，几何模型都作为散布像素渲染到显示屏上，其中一些可以交互地更改图片。

Digital Light dawned between the American A-bomb of 1945 and the Soviet H-bomb of 1949—between World War II and the Cold War. Recall from the Dawn of Digital Light chapter that making pictures on the few precious computers in the 1940s was considered frivolous. They “should” be computing A-bomb or H-bomb statistics. Nevertheless, the first digital picture, First Light, appeared in 1947 on Manchester’s Baby. Two of the first documented interactive videogames were created in 1951 and 1952 at Manchester and Cambridge Universities. And the first documented computer animations appeared on Whirlwind at MIT in 1951, funded by the Air Force. In all these early cases, geometric models were rendered to a display screen as spread pixels, and some of them could change pictures interactively. Whirlwind is at the top of this chapter’s flow chart because it connects that early history to this Soviet-fueled, defense-funded next stage—one nevertheless not devoid of occasional frivolous picture making.

冷战催生了 1950 年代最冷血的野兽，或者更确切地说，这些野兽肩负着最冷血的使命——防御核攻击。当时最大、最可怕的是 Sage 计算机，它是旋风的直接继任者。空军与麻省理工学院和 IBM 合作实施 Sage，该项目的规模和成本可与曼哈顿或阿波罗计划相媲美。从 1958 年开始，建造了 20 多个中心，每个中心都有四层楼高，没有窗户，并装有两台 Sage 计算机。每台计算机造价 2.38 亿美元（约合今天的 20 亿美元），占地半英亩占地面积，重达100多吨。每个都有一个通常是黄色的图形显示器，带有一个用于交互的“光枪”。

The Cold War spawned the most cold-blooded beasts of the 1950s, or rather beasts with the most cold-blooded mission—defense against nuclear attack. The biggest and scariest at the time was the Sage computer, the immediate successor to Whirlwind. The Air Force worked with MIT and IBM to implement Sage, a project comparable in size and cost to the Manhattan or Apollo Projects. Starting in 1958 more than 20 centers were built, each four stories tall, windowless, and housing two Sage computers. Each computer cost $238 million (about $2 billion in today’s dollars), occupied half an acre of floor space, and weighed over 100 tons. And each featured an often-yellow graphical display with a “light gun” for interaction.

今天，Sage 剩下的文物包括那些图形站。图 6.14（右）中的军事用户，凝视着野兽张开的大嘴，似乎准备射击它——或者无论如何都要焊接它。为了对这种努力有一个正确的认识，想象一下一个房间里挤满了数十名军人和他们的枪，在排列整齐的展示站。

Among the remaining artifacts of Sage today are those graphics stations. The military user in figure 6.14 (right), staring into the gaping maw of the beast, appears ready to shoot it—or solder it anyway. To get a proper sense of this effort, picture a room filled with dozens of servicemen and their guns at display stations arranged in rank and file.

Sage 的正常业务端（图 6.15）描绘了飞机和导弹在粗糙边界线上的位置。显示右侧的虚线——一个“不明飞行物”——接近美国。但《大西洋月刊》在一篇题为“世界上第一个计算机艺术（这是一个性感的贵妇人）的前所未闻的故事”的文章中讲述了另一个故事。有人，当然是匿名的，使用价值 2.38 亿美元的机器来展示“禁果”——来自Esquire的少女海报，男性杂志。这篇文章声称，她是 1950 年代后期的某个时候，第一次在电脑显示器上描绘了一个人。这甚至可能是真的，但请记住第 4 章提到的可能在 1950 年代初在 Edsac 上的高地舞者的轶事。无论如何，这些图形是已经灭绝的书法或矢量品种，是失败者从长远来看，像素的光栅显示，现代形式。尽管 Sage 恐龙的体型很大，但它的图形很简单——但它们是互动的。24

The normal business end of Sage (figure 6.15) depicts aircraft and missile locations against crude borderlines. A dotted line at display right—an “unidentified flying object”—approaches the United States. But the Atlantic Monthly told an alternative story in an article titled “The Never-Before-Told-Story of the World’s First Computer Art (It’s a Sexy Dame).” Someone, anonymous of course, used the $238 million machine to display “forbidden fruit”—a girlie pinup traced from Esquire, the men’s magazine. The article claims that she was the first depiction of a person on a computer display, sometime in the late 1950s. This might even be true, but remember that anecdote of a Highland dancer on Edsac, probably in the early 1950s, mentioned in chapter 4. In any case these graphics were the calligraphic, or vector, variety that have come and gone extinct, losers in the long run to raster displays of pixels, the modern form. Despite the size of the Sage dinosaur, its graphics were simple—but they were interactive.24

我们所说的实时或交互式是什么意思？

What Do We Mean by Real Time—or by Interactive?

“实时”一词有多种含义，都与人类对时间流逝的感知有关。一种流行的含义是实时计时，就像在电子游戏中一样。其他是实时按需的，就像在计算机模型的“交互式”设计中一样。这些方便的术语可以帮助我们细分数字光的广阔范围。

The term “real time” has several meanings, all related to the human perception of the passage of time. One popular meaning is clocked real time, as in videogames. Another is real time on demand, as in the “interactive” design of computer models. These are handy terms that help us subdivide the vastness of Digital Light.

在电子游戏中，一个假想的时钟滴答作响，并且必须渲染一张新图片并准备好在每次滴答时显示。视频的自然“时钟”是每秒 30 帧的帧速率，之所以选择这个速率，是因为它近似于人类对时间流逝的感觉，并且与现实世界中的时间同步。因此，在电子游戏中，每三十秒必须显示一张新图片。

In videogames, an imaginary clock ticks, and a new picture must be rendered and ready for display at each tick. The natural “clock” for video is the frame rate of 30 frames per second, a rate chosen because it approximates the human sense of time passing smoothly and at the same pace as time in the real world. In videogames, then, a new picture must be displayed every thirtieth of a second.

电子游戏是现代世界时钟品种实时图形的压倒性例子。但在早期，飞行模拟器——每台花费数百万美元——是主要表现形式。战斗机飞行员将在飞行模拟器上进行训练，而宇航员将在航天器模拟器上进行训练。他们今天仍然这样做。虚拟现实 (VR) 是时钟实时类的最新成员，可在特殊护目镜内渲染到双显示器，以获得立体或所谓的 3D 效果。

Videogames are the overwhelming example of real-time graphics of the clocked variety in the modern world. But in the early days, flight simulators—costing millions of dollars each—were the principal expression. Fighter pilots would train on flight simulators, and astronauts on spacecraft simulators. They still do today. Virtual reality (VR) is the newest member of the clocked real-time class, rendering to dual displays inside special goggles for a stereo, or so-called 3D, effect.

非时钟或按需实时图形的一个主要示例是内部计算机模型的交互式设计。这只是另一种说法，“图片变化得足够快，足以让用户相信它在与计算机交互时‘即时’发生。”

A principal example of unclocked, or on-demand, real-time graphics is interactive design of internal computer models. That’s just another way of saying “pictures that change fast enough to convince users that it’s happening ‘instantaneously’ while they interact with the computer.”

另一个例子是几乎每个应用程序的每个界面。我只是单击了 Word 中的保存图标。当我将鼠标光标悬停在它上面时，图标“立即”亮起，然后当我单击鼠标并保存文件时，“时间正在消逝”的旋转图标“立即”出现。

Another example is almost every interface to every app. I just clicked on the Save icon in Word. The icon “instantly” lit up as I hovered the mouse cursor over it, then a “time-is-elapsing” rotating icon appeared “instantly” as I clicked the mouse and saved the file.

在时钟或非时钟情况下，实时的权衡要么是低图像质量，要么是高产品价格。在时钟的情况下，电子游戏仍然（截至 2020 年）显示在保持低消费价格的同时，没有足够的时间来计算完全渲染的图片的视觉伪影。

In either the clocked or unclocked case, the trade-off for real time is either low image quality or high product price. In the clocked case, videogames still (as of 2020) show the visual artifacts of insufficient time to compute a fully rendered picture while maintaining a low consumer price.

在非时钟情况下，模型交互设计中使用的图片质量通常仍低于如此设计的对象或场景的最终渲染。设计师可能会观看他们正在创建的对象或角色的线描显示 - 例如茶壶的金属丝网视图，而不是完全着色、彩色和纹理的茶壶。

In the unclocked case, the picture quality used in interactive design of models is still typically lower than the final rendering of the objects or scenes so designed. Designers might watch a line-drawing display of an object or character that they are creating—say a wire mesh view of a teapot rather than a fully shaded, colored, and textured teapot.

计算机图形学的公认历史

The Received History of Computer Graphics

计算机图形学是我的职业，但我对它的历史只有一个摇摇欲坠的把握。我收到的版本大致是这样的：Ivan Sutherland 于 1962 年在麻省理工学院编写了第一个交互式计算机图形程序 Sketchpad。然后他于 1968 年进入盐湖城的犹他大学，开创了计算机图形学的王朝先驱者。事实上，这个总结有很多道理。例如，Ed Catmull 是 Sutherland 的学生之一，多年来我们在纽约理工学院、卢卡斯影业和皮克斯的许多同事都在犹他州接受过教育。我们使用的早期图形硬件是由 Evans & Sutherland 制造的，这家公司也是 David Evans 和 Ivan Sutherland 于 1968 年在盐湖城共同创立的。

Computer graphics is my profession, but I only had a shaky hold on its history. The version I received was roughly this: Ivan Sutherland wrote Sketchpad, the first interactive computer graphics program, it was said, at MIT in 1962. Then he went to the University of Utah in Salt Lake City in 1968 and fathered a dynasty of computer graphics pioneers. In fact, there’s a lot of truth in that summary. For example, Ed Catmull was one of Sutherland’s students, and many of our colleagues over the years at the New York Institute of Technology, Lucasfilm, and Pixar were Utah educated. And the early graphics hardware we used was manufactured by Evans & Sutherland, a company David Evans and Ivan Sutherland cofounded in Salt Lake, also in 1968.

但萨瑟兰的叙述与我们在“数码光之黎明”一章中所发现的不符：第一张数码图片、互动游戏和动画是在 1940 年代末和 1950 年代初出现在最早的计算机上的。Whirlwind 有一个光枪，可以让你与它的显示器互动。正如我们在本章中看到的那样，在 1950 年代后期，正在射击或焊接 Sage 的人正在与它的图形进行交互。Spacewar在 Sketchpad 之前就在麻省理工学院进行了互动。显然，故事中存在差异需要解释。

But the Sutherland narrative doesn’t fit what we’ve discovered in the Dawn of Digital Light chapter: The first digital pictures, interactive games, and animations were on the earliest computers in the late 1940s and early 1950s. Whirlwind had a light gun that let you interact with its display. As we’ve seen in this chapter, the fellow who was shooting, or soldering, Sage is interacting with its graphics in the late 1950s. And Spacewar was interactive at MIT slightly before Sketchpad. Obviously, there’s a discrepancy in the story that needs explanation.

像许多技术史一样，收到的版本省略了很多内容，并且过分强调了一个人。它忽略了该领域的两位著名创始人史蒂文·库恩斯（Steven Coons）和皮埃尔·贝塞尔（Pierre Bézier），他们比萨瑟兰早了一代人。然而，该领域非常了解这些创始人。几十年来，计算机图形学的最高奖项一直是 Coons 奖。几十年来，流行的应用程序 Adob​​e Illustrator 和 Adob​​e Photoshop 也采用了贝塞尔曲线。但是历史网络并没有顺利编织在一起。

Like many histories of technology, the received version omits great swaths, and overly emphasizes a single person. And it leaves out two well-known founding fathers of the field, Steven Coons and Pierre Bézier, who predated Sutherland by a generation. The field is quite aware of these founders, however. For decades, the highest award in computer graphics has been the Coons Award. And Bézier curves have been featured in the popular apps Adobe Illustrator and Adobe Photoshop for decades as well. But there’s been no smooth weaving together of the historical web.

浣熊和贝塞尔

Coons and Bézier

将 Coons 和 Bézier 称为计算机图形学的奠基人的问题在于，他们都没有制作图片，至少一开始也没有。Coons 在旧的 Chance Vought 公司创建了他在飞机行业的基础工作。贝塞尔在法国汽车制造商雷诺完成了他的工作。他们都在计算机内部使用了三维表面的几何模型，但都没有显示设备来查看图片。Coons 创造了实际的机身表面和 Bézier 实际的汽车车身表面。

The problem with calling Coons and Bézier the founding fathers of computer graphics is that neither made pictures, not at first, anyway. Coons created his foundational work in the aircraft industry at the old Chance Vought company. Bézier did his at Renault, the French automobile manufacturer. They both used geometric models of three-dimensional surfaces inside a computer, but neither had a display device to see pictures. Coons created actual airframe surfaces, and Bézier actual auto body surfaces.

他们并不关心图片本身。他们关心由真实材料制成的真实 3D 空间中的真实物体——汽车和飞机机身，或它们的模型。这又是创意空间与展示空间的区别。Coons 和 Bézier 有创意空间，但没有展示空间。或者换句话说，他们的展示是物体本身，而不是它们的照片。他们拥有计算机控制的机器，可以从真实材料中雕刻或切割实际物体。换句话说，他们是针对对象进行计算机辅助设计，而不是针对图片进行计算机图形设计。我们将在本章后面花时间解开这些领域。但毫无疑问，Coons 和 Bézier 都是两者的主张。他们是一个很好的起点。

They didn’t care about pictures per se. They cared about actual objects in real three-dimensional space made of real materials—automobile and airplane bodies, or maquettes for them. It’s the Creative Space versus Display Space distinction again. Coons and Bézier had Creative Space but no Display Space. Or to put it another way, their displays were the objects themselves, not pictures of them. They had computer-controlled machines that carved or cut actual objects from real materials. In other words, they were doing computer-aided design aimed at objects, not computer graphics aimed at pictures. We’ll spend time later in this chapter disentangling these fields. But there’s no question that Coons and Bézier are claimed by both. They are a good place to start.

被遗忘的计算机图形学历史：三驾马车

The Forgotten History of Computer Graphics: The Triumvirate

黎明一章的一个教训是，数码光始于 1947 年，并在 1950 年代初期蓬勃发展——包括计算机图形学，数码光的一个分支，它将内部几何模型渲染成可见的图片。计算机图形学的早期十年，就像 Coons 和 Bézier 早期的 CAD 工作一样，几乎从未被提及。相反，该领域的通常历史始于 Sketchpad。我们的部分工作是阐明是什么让 Sketchpad 如此重要。

A lesson from the Dawn chapter is that Digital Light began in 1947 and flourished in the early 1950s—including computer graphics, the branch of the Digital Light that renders internal geometric models into visible pictures. That early decade of computer graphics, like the earlier CAD work of Coons and Bézier, is hardly ever mentioned. The usual history of the field starts, instead, with Sketchpad. Part of our job is to articulate what made Sketchpad so important.

Ivan Edward Sutherland 于 1962 年在麻省理工学院展示了著名的 Sketchpad，并完成了它并于 1963 年 1 月作为他的博士论文发表。这是一个以用户交互为特色的二维几何设计程序。它当时受到高度吹捧，现在仍然是——而且应该是。但计算机图形学的荣耀是在二维透视图中查看三维场景和物体。52

Ivan Edward Sutherland famously demonstrated Sketchpad in 1962 at MIT and completed it and published it in January 1963 as his PhD thesis. It was a two-dimensional geometrical design program that featured user interaction. It was highly touted then and still is—and should be. But the glory of computer graphics is three-dimensional scenes and objects, viewed in two-dimensional perspective.52

是 Sutherland 的研究生 Timothy Edward “Tim” Johnson 将 Sketchpad 放入了三个维度和透视图。他将他的程序命名为 Sketchpad III，并于 1963 年 5 月发表了他的硕士论文。“III”是三个尺寸，而不是第三版。（没有 Sketchpad II。）为什么 Sketchpad III 没有以同等的力量被记住？可以说，从二维转换为三维并没有那么大的变化，实际上是一个相当明显的变化。但观点是巨大的、不平凡的、不明显的。Sketchpad III 采用了 Johnson 从另一位研究生 Lawrence Gilman “Larry” Roberts 那里学到的透视解决方案。罗伯茨于 1963 年 6 月完成了他的博士论文。因此，1963 年是计算机图形学三巨头的一年，所有这些都在麻省理工学院的林肯实验室：伊万·萨瑟兰、蒂姆·约翰逊和拉里·罗伯茨。两个，萨瑟兰和约翰逊，是史蒂文库恩斯的学生。

It was Sutherland’s fellow graduate student Timothy Edward “Tim” Johnson who put Sketchpad into three dimensions and perspective. He called his program Sketchpad III and published his master’s thesis on it in May 1963. The “III” was for three dimensions, not third version. (There was no Sketchpad II.) Why isn’t Sketchpad III remembered with equal force? One can argue that converting from two dimensions to three wasn’t that big a change, indeed a rather obvious one. But perspective was huge, nontrivial, and nonobvious. Sketchpad III incorporated the solution for perspective that Johnson learned from yet another fellow graduate student, Lawrence Gilman “Larry” Roberts. Roberts completed his PhD thesis in June 1963. So 1963 was the year of the Triumvirate of computer graphics, all at MIT’s Lincoln Lab: Ivan Sutherland, Tim Johnson, and Larry Roberts. And two, Sutherland and Johnson, were students of Steven Coons.

图 6.28 显示了代表三个各自成就的图标，Sutherland 画板的正方形，Johnson 画板 III 的立方体，以及 Roberts 对 Sketchpad III 贡献的透视立方体。

Figure 6.28 shows icons representing the three respective achievements, the square for Sutherland’s Sketchpad, the cube for Johnson’s Sketchpad III, and the cube in perspective for Roberts’s contribution to Sketchpad III.

图 6.29 以相同的顺序显示了 MIT 同学三巨头的照片：萨瑟兰、约翰逊、罗伯茨。53

And figure 6.29 displays photos of the Triumvirate of MIT classmates in the same order: Sutherland, Johnson, Roberts.53

计算机图形学的中心法则

The Central Dogma of Computer Graphics

以下是如何看待当今的计算机图形，无论它在哪里使用——电影、游戏、VR 等等：一个虚构的世界是在计算机内存中建模的。这是模型，不是图片。它是标准的三维几何 á la Euclid 和其他早期希腊人。它不可见。这个世界的图像是二维的。所以为了拍一张照片，一个虚构的相机被放置在虚构的世界中。摄像头是虚拟的。它的机身、镜头、三脚架等没有建模。我们只关心虚拟摄像机在场景中的位置、指向的方向以及向上的方向。就像一个真正的相机一样，这个虚拟相机有一个取景器，可以捕捉到它所看到的虚构世界的矩形视图。我们称取景器为透视视口 在计算机图形学中——只是viewport，简称。Tim Johnson 首先在 Sketchpad III 中实现了（透视）视口。56

Here’s how to think about computer graphics today, regardless of where it’s used—in movies, games, VR, or so forth: A fictitious world is modeled inside a computer memory. It’s a model, not a picture. It’s standard three-dimensional geometry á la Euclid and other early Greeks. It’s not visible. A picture of this world is two-dimensional. So to make a picture, a fictitious camera is placed in the fictitious world. The camera is virtual. Its body, lenses, tripod, and so forth are not modeled. All we care about is where the virtual camera is located in the scene, what direction it’s pointed, and which way is up. Like a real camera, this virtual one has a viewfinder that captures a rectangular view of the fictitious world it’s looking at. We call the viewfinder a perspective viewport in computer graphics—just viewport, for short. Tim Johnson first implemented a (perspective) viewport in Sketchpad III.56

虚拟相机在其视口中捕获的虚幻世界部分被展平为二维 - 就像真实相机在其取景器中将现实世界展平为二维图像一样。还有一件非常重要的事情：视口以透视的方式看到了虚幻的世界——就像真实相机的取景器一样。这与我们人类感知视觉世界的视角相同。Johnson 在 Sketchpad III 的视口中使用了 Larry Roberts 的透视解决方案。

The portion of the unreal world that the virtual camera captures in its viewport is flattened into two dimensions—just like a real camera flattens the real world into a two-dimensional image in its viewfinder. And one further, very important, thing: the viewport sees the unreal world in perspective—just like a real camera’s viewfinder does. It’s the same perspective through which we humans perceive the visual world. Johnson used Larry Roberts’s perspective solution in the viewport of Sketchpad III.

我称其为计算机图形学的中心法则，因为它不必如此。毕竟，计算机是人类发明的最具延展性的工具。几何不一定是欧几里得，但它在中心法则中。它可能是科捷利尼科夫的祖父所鼓吹的洛巴乔夫斯基式——没有平行线（见第 2 章）。最重要的是，视角不一定是文艺复兴时期的视角，但中央教条就是这样。简而言之，中心法则代表了一个不真实的世界，就像我们人类感知的真实世界一样。这是虚拟相机对虚幻世界的透视图，全部在计算机内部，最终被渲染成像素，以便我们可以实际看到它。

I call this the Central Dogma of computer graphics because it doesn’t have to be that way. After all, a computer is the most malleable instrument ever invented by humankind. The geometry doesn’t have to be Euclidean, but it is in the Central Dogma. It could be Lobachevskian—with no parallel lines—as trumpeted by Kotelnikov’s grandfather (see chapter 2). Most importantly the perspective doesn’t have to be Renaissance perspective, but the Central Dogma has it so. In short, the Central Dogma represents an unreal world exactly as we humans perceive the real world. It’s the virtual camera’s perspectival view of the unreal world, all inside the computer, that is finally rendered into pixels so that we can actually see it.

MC Escher 的艺术展示了扭曲的视角会变得多么糟糕。在图 6.31 中，受 Escher 的启发，在没有计算机的情况下完成，直梁显然必须弯曲，以使它们的网格在透视中消失，朝向无限远的两点。这两个消失点在三维空间中彼此垂直的线上，但它们在一个平面中沿着一条线出现。这在透视图中并不罕见。但是这幅画的目的是让从下消失点发出的扩散线在看向上消失点时变得明显不断地间隔开。需求如此之大，以至于从较低的消失点朝向观察者的直线必须向上弯曲以适应它。而那些来自上消失点的人必须向下弯曲。这是非人的观点。我们看不到直线弯曲，中央教条也不允许。（这张图片，来自中心教条之外，被使用了 38 年作为年会的封面计算机科学程序的基础。）

The art of M. C. Escher shows just how screwy perspective can get. In figure 6.31, inspired by Escher and done without a computer, the straight beams have to bend, apparently, to let a grid of them vanish in perspective toward two points at infinity. The two vanishing points are on lines perpendicular to one another in three-dimensional space, but they appear along a single line in one plane. That’s not unusual in perspective drawings. But the picture’s intent is to have spreading lines that emanate from the lower vanishing point become apparently constantly spaced as seen looking toward the upper vanishing point. The demand is so severe that straight lines coming toward the viewer from the lower vanishing point have to bend upward to accommodate it. And those coming from the upper vanishing point have to bend downward. That’s a nonhuman perspective. We don’t see straight lines bend, and the Central Dogma doesn’t allow it. (This picture, from outside the Central Dogma, was used for 38 years as the cover for the annual Foundations of Computer Science proceedings.)

计算机辅助设计中计算机图形学的早期起源，例如，在 Coons、Bézier 和 de Casteljau 的工作中，促成了中心教条。在 CAD 中，设计的对象必须存在于现实世界中，这是必不可少的。但是，尽管中心法则是几乎总是在计算机图形学中实践的学科，但它肯定不是必需的。计算机艺术家从一开始就违反了它。从某种意义上说，这就是他们作为艺术家的角色——在没有人为限制的情况下运用数字光的媒介，并向我们其他人展示他们的发现。但是有即使在中心教条学科中也有很多创造力。拍摄皮克斯、梦工厂、蓝天和其他制作动画故事片的电影。学科内有如此多的创意空间，以至于许多艺术家从不考虑在它之外冒险。

The early origins of computer graphics in computer-aided design, in the work of Coons, Bézier, and de Casteljau, for instance, contributed to the Central Dogma. It’s a necessity in CAD where the objects designed must exist in the real world. But although the Central Dogma is the discipline that’s nearly always practiced in computer graphics, it’s certainly not required. Computer artists have violated it since the beginning. In a sense, that’s their role as artists—to exercise the medium of Digital Light without artificial limits and show the rest of us what they find. But there’s been a lot of creativity even within the Central Dogma discipline. Take any movie by Pixar, DreamWorks, Blue Sky, and others creating animated feature films. There’s so much Creative Space within the discipline that many artists never consider venturing outside it.

这是妙语。1963 年，麻省理工学院的约翰逊和罗伯茨向我们介绍了现在使用的中心法则（尽管他们没有这么称呼它）。他们应该得到比他们往往得到的更多的荣誉。特别是，他们获得的信用比萨瑟兰少。但在某种程度上，这三个人都参与其中。这就是为什么我称他们为三巨头。在扩展这三位领导者之前，让我们简要回顾一下计算机图形学中透视的历史，因为约翰逊和罗伯茨并不是第一个使用透视的人。他们也不是首先使用中央教条。

Here’s the punch line. Johnson and Roberts at MIT in 1963 were the ones who introduced us to the Central Dogma as it’s now used (although they didn’t call it that). They deserve more credit than they tend to get. In particular, they get less credit than Sutherland. But all three were in on it at some level. That’s why I call them the Triumvirate. Before expanding on these three leaders, let’s briefly review a history of perspective in computer graphics, because Johnson and Roberts weren’t the first with perspective. Nor were they first with the Central Dogma.

伊万究竟做了什么？

What Exactly Did Ivan Do?

像 Steven Coons 一样，Ivan Sutherland 这个名字在计算机图形学中一次又一次地出现。我的同事 Ed Catmull 在 Sutherland 手下学习，多年来向我讲述了他。我什至和萨瑟兰在同一个房间里去过几次，但我从来没有真正了解过这个人。因此，2017 年 5 月 9 日，我在俄勒冈州波特兰市的我位于英格兰剑桥的公寓里与他进行了 Skype 通话——根据他的要求，他的时间是早上 5 点，这是他开始新一天的通常时间。

Like Steven Coons, the name Ivan Sutherland shows up again and again in computer graphics. My colleague Ed Catmull studied under Sutherland and told me about him over the years. I had even been in the same room as Sutherland a couple of times, but I’d never really gotten to know the man. So, on May 9, 2017, I Skyped him in Portland, Oregon, from my flat in Cambridge, England—at 5 a.m. his time per his request, the usual time he starts his days.

我们一起度过了一个半小时，笑声比我预期的要频繁得多。萨瑟兰并不是我所期望的那种严厉的人。是的，他最初的举止暗示了一种阴郁的性格，但我很快就发现了一丝微笑，几乎是隐藏的。偶尔会出现一些诙谐的评论——带着有趣的一口。他对我的开场白提醒我，Sketchpad 的发展发生在笨重的恐龙计算机时代，这个时代充满了电子塔。他告诉我，他必须步行穿过TX-2 才能在 Sketchpad 上工作。

We spent an hour and a half together, chuckling much more often than I had anticipated. Sutherland was not the stern man I expected from reports. Yes, his initial demeanor suggested a saturnine personality, but I soon detected a slight smile, almost hidden. And there was the occasional witty remark—with a funny bite to it. His opening salvo to me was a reminder that Sketchpad’s development occurred in the era of hulking dinosaur computers, this one filling a room with towers of electronics. He had to walk, he told me, through TX-2 to work on Sketchpad.

麻省理工学院的 TX-2 可能是当时世界上最强大的单用户计算机。它的家是林肯实验室，一个高度安全的秘密设施（再次，一个军事暴君提供掩护）。但作为林肯实验室的员工和麻省理工学院的学生，萨瑟兰可以接触到这只野兽。他必须自己使用它，就好像它是他自己的一样。满屋子的电子设备只对他有反应。TX-2 是他自己的私人霸王龙，带有图形显示器和光笔。萨瑟兰可以用笔操纵显示器上的点。

TX-2 at MIT was probably the world’s most powerful single-user computer at the time. Its home was Lincoln Lab, a secret facility with high security (once more, a military tyrant offers cover). But as a Lincoln Lab employee and MIT student, Sutherland had access to the beast. And he got to use it by himself as if it were his own. The whole roomful of electronics responded only to him. TX-2 was his own personal T Rex, with a graphics display and a light pen. Sutherland could manipulate points on that display with the pen.

他将线段模型存储在 TX-2 的内存中，并编写了一个程序，将这些不可见的二维模型渲染为可见的二维TX-2 的光栅书法杂交显示器上的图片。如果他让光笔悬停在显示器上的某个点附近——比如线段的末端——那么他可以抓住端点并将其移动到新位置。存储在内存内部模型中的端点坐标会发生变化。显示屏将显示更改模型的更新图片。

He stored models of line segments in the memory of TX-2 and he wrote a program that rendered these invisible two-dimensional models into visible two-dimensional pictures on TX-2’s raster-calligraphic crossbreed display. If he let the light pen hover in the vicinity of a point on the display—say the end of a line segment—then he could grab the endpoint and move it to a new location. The endpoint coordinates stored in the internal model in memory would change. And the display would show an updated picture of the changed model.

萨瑟兰还首次展示了橡皮筋。他会抓住线段的一端并将其拖到新位置。旧线段消失，因为新线段被立即绘制到新位置。因此，随着他移动光笔，线段被重新绘制。它会随着新端点的需要和用户指示的任何方向延长或缩短。该技术及其效果令观众惊叹不已。现在看起来很明显，但以前没有人见过。

Sutherland also showed rubberbanding for the first time. He would grab one end of a line segment and drag it to a new location. The old line segment disappeared as a new line segment was drawn instantaneously to the new location. So the line segment was redrawn as he moved the light pen. It would lengthen or shorten as the new endpoint required and in whatever direction the user indicated. The technique and its effects amazed viewers. It seems so obvious now, but nobody then had seen it before.

Sutherland 的二维 Sketchpad 强调了几何约束的力量以及对他的模型进行应力分析和其他物理测试的能力。这使 Sutherland 直接进入了计算机辅助设计开发的主线。

Sutherland’s two-dimensional Sketchpad emphasized the power of constraints on geometry and the ability to do stress analysis and other physical tests on his models. This put Sutherland directly in the main line of computer-aided design development.

麻省理工学院三巨头的另一位成员蒂姆·约翰逊（Tim Johnson）对我来说，可以这么说，萨瑟兰是这样看待的。我在下面的故事中保留了工程（计算机辅助设计）的讨论，这也让我们想起了 Coons 的影响：

Tim Johnson, another member of the MIT Triumvirate, put Sutherland into perspective, so to speak, for me. I’ve left the engineering (computer-aided design) talk intact in the following story, which also reminds us of Coons’s influence:

史蒂夫库恩斯出现了一晚。. . 第一次看到 Ivan 的图形约束在起作用。Coons 被“恒定长度”约束收回，该约束固定了行长。他 [库恩斯] 随便注意到。. . 他可以绘制一个桁架并显示应变，只需为每条受约束的线添加一个文本显示，以显示该线正在经历的拉伸/压缩量。伊万从来没有预料到会有这样的用途。他为下一个演示添加了文本，史蒂夫开始绘制经典结构来测试它。那一刻标志着伊万工作的巨大力量。65

Steve Coons showed up one night . . . to see Ivan’s graphical constraints in action for the first time. Coons was taken back by the “constant length” constraint, which fixed a line length. He [Coons] casually noted . . . that he could draw a truss and show the strain just by adding a text display for each constrained line that showed the amount of stretch/compression the line was undergoing. Ivan had never anticipated such a use. He added the text for the next demo and Steve went to work drawing classic structures to test it. That moment marked the immense power of Ivan’s work.65

抛开限制不谈，Sketchpad 中的图形显示水平按照今天的标准似乎几乎是微不足道的，但在 1962 年没有人见过这样的东西。或者他们见过吗？像往常一样，一项技术的历史并不像收到的版本那么简单。与此同时，还有另一个交互式计算机图形和计算机辅助设计程序正在开发中。计算机增强设计 (DAC-1) 是在 IBM 的帮助下创建的，仅供底特律通用汽车公司内部的设计师和工程师使用。它的合同已出租（授予的法律术语）) 于 1960 年在纽约金斯敦的 IBM 进行开发，直到 1963 年 4 月 IBM 将该系统交付给 GM。它始终是一个三维系统。GM 知道 Sketchpad，但认为二维线段不足以满足其设计目的。他们需要三维曲线。66

Constraints aside, the level of graphic display in Sketchpad seems almost trivial by today’s standards, but nobody had seen anything like it in 1962. Or had they? As usual the history of a technology isn’t as simple as the received version. There was another interactive computer graphics and computer-aided design program in development at the same time. Design Augmented by Computers (DAC-1) was created, with IBM’s help, for exclusive use by designers and engineers internal to General Motors in Detroit. The contract for it was let (legalese for awarded) in 1960 and development occurred at IBM in Kingston, New York, until IBM delivered the system to GM in April 1963. It was always a three-dimensional system. GM was aware of Sketchpad but considered two-dimensional line segments insufficient for its design purposes. They needed three-dimensional curves.66

起初并没有推动系统公开。也就是说，直到 1963 年 Sketchpad 和 Sketchpad III 出版。通用汽车很快得到了 1964 年出版的 DAC-1 来分享在荣耀中。但很少有人记得它或它的作者 Edwin Jacks、Barrett Hargreaves 和许多其他人。67

At first there was no push to make the system public. That is, not until Sketchpad and Sketchpad III published in 1963. GM quickly got DAC-1 published in 1964 to share in the glory. But few remember it or its authors, Edwin Jacks, Barrett Hargreaves, and many others.67

在我准备本章时，Barrett Hargreaves 自己告诉我：

Here’s what Barrett Hargreaves himself told me as I prepared this chapter:

是的，您可以使用光笔输入线段。设计师可以输入两个位置并要求它们之间有一条直线。或者有一些计算机图形对话程序可以让他开发曲线。例如，他可以输入现有的 3-D 表面，并将线或表面与该表面或与原始表面的给定距离相匹配。最终结果可能是驱动数控机床的磁带。

DAC-1 系统是一个非常早期的开发，它聘请了工程师和设计师与 GM Research 的计算机专家和数学家一起工作。工程师告诉我们他们需要什么，我们对他们的要求进行编程响应。工程师确实使用 DAC-1 来设计车辆部件，主要是引擎盖和挡泥板等车身部件。

DAC-1 对 GM 最重要的贡献可能是，在 1960 年代初期，它向 GM 工程师和 GM 工程管理人员证明了计算机图形可以有效地用于设计和工程。计算机用于燃油经济性计算、车辆操纵计算、凸轮应力分析等的案例很多，但 DAC-1 证明，使用计算机可以极大地协助车身部件和其他面向图形的工程。68

Yes, you could use the light pen to enter line segments. The designer could enter two locations and request a straight line between them. Or there were a number of computer graphic dialog routines that allowed him to develop curved lines. He could for instance, enter an existing 3-D surface and match a line or surface to that surface or a given distance from the original surface. The end result could be a tape that drove a numerically controlled machine tool.

The DAC-1 system was a very early development that employed engineers and designers working along with GM Research computer specialists and mathematicians. The engineers told us what they needed and we programmed responses to their requests. The engineers did use DAC-1 to work on the design of vehicle parts, mostly body parts like hoods and fenders.

Probably the most important contribution of DAC-1 to GM was that, in the very early 1960s, it demonstrated to GM engineers and GM engineering management that computer graphics could effectively be used in design and engineering. There were numerous cases where computers were used for fuel economy calculations, vehicle handling calculations, cam stress analysis, etc. but DAC-1 proved that body parts and other graphic oriented engineering could be greatly assisted using the computer.68

根据哈格里夫斯的说法，通用汽车与麻省理工学院合作开发了 DAC-1 和光笔的软件。麻省理工学院和通用汽车集团会互相了解。这里有一些涉及 DAC-1 的有趣历史线索，值得进一步调查。DAC-1 帐户中缺少的一件事是确切的日期。比如，画板只有二维线条图的时候，有没有三维曲线？在约翰逊的画板 III 之前它有 3D 交互图形吗？DAC-1 于 1963 年 4 月的交付早于 Sketchpad III 于 1963 年 5 月的出版，这表明它确实如此，但这些都是外部日期，不一定表示内部发展。我们确实知道 DAC-1 没有透视显示。69

According to Hargreaves, GM worked with MIT on software for DAC-1 and the light pen. The MIT and GM groups would have known about one another. There are intriguing historical hints here involving DAC-1 that deserve further sleuthing. One thing lacking in DAC-1 accounts is exact dates. For example, did it have three-dimensional curves when Sketchpad had only two-dimensional line drawings? Did it have three-dimensional interactive graphics before Johnson’s Sketchpad III? That DAC-1’s delivery in April 1963 predates Sketchpad III’s publication in May 1963 suggests that it did, but these are both external dates and not necessarily indicative of internal developments. We do know that DAC-1 did not display in perspective.69

哈格里夫斯还提供了以下惊喜——通用汽车公司的早期计算机动画的证据显示给沃尔特迪斯尼本人，但不是在 DAC-1 上：

Hargreaves also delivered the following surprise—evidence of an early computer animation at GM shown to Walt Disney himself, but not on DAC-1:

沃尔特迪斯尼在 1960 年初的某个时间访问了通用汽车研究实验室计算机技术部。一位名叫 Hugh Brouse 的计算机图形程序员同事为迪斯尼先生开发了一部米老鼠的自动化电影。迪斯尼先生着迷了。当他在我们小小的计算机部门走来走去时，我们都目瞪口呆。70

Walt Disney visited the GM Research Laboratories Computer Technology Department sometime in the early 1960 time frame. A fellow computer graphics programmer named Hugh Brouse had developed an automated movie of Mickey Mouse for Mr. Disney. Mr. Disney was fascinated. We all stood agape as he walked around our little computer department.70

同样，日期很柔和，缺乏细节。休·布鲁斯去世了。因此，如果没有确凿的证据证明并非如此，并使用我们保守的约会计划来确定优先级，让我们假设此后在 Sketchpad 与 DAC-1 的比赛中，萨瑟兰是第一名。毫无疑问，他的 Sketchpad 广为人知，而 DAC-1 却没有。但同样清楚的是，DAC-1 几乎同时是一个交互式计算机辅助设计程序。71

Again, the dates are soft and details lacking. And Hugh Brouse has died. So, without hard evidence that proves otherwise, and using our conservative dating scheme for establishing priority, let’s assume henceforth in the Sketchpad versus DAC-1 contest that Sutherland was first. It’s certainly true that his Sketchpad was publicized far and wide, and DAC-1 was not. But it’s also clear that DAC-1 was an interactive computer-aided design program at about the same time.71

但是，如果我们给他，萨瑟兰首先做了什么？它不是交互式计算机图形。十年前，克里斯托弗·斯特拉奇（Christopher Strachey）和他的草稿（跳棋）电子游戏以及桑迪·道格拉斯（Sandy Douglas）和他的三字棋（井字棋）电子游戏已经做到了这一点。1950 年代早期的这两个程序都允许用户交互地更改，例如，将一个空的棋盘正方形更改为一个持有 X 或 O 的棋盘。更新了内部棋盘模型，并重新绘制了显示以显示新棋盘配置锯齿状地呈现 Xs 和 Os。

But what exactly had Sutherland done first, if we give him that? It wasn’t interactive computer graphics. That had already been done a decade earlier—by Christopher Strachey and his draughts (checkers) videogame and by Sandy Douglas and his noughts and crosses (tic-tac-toe) videogame. Both programs from the early 1950s enabled a user to interactively change, say, a board square from an empty one to one holding an X or an O. The internal board model was updated, and the display was redrawn to show the new board configuration with jaggedly rendered Xs and Os.

这确实是 Sutherland 的创新：Sketchpad 交互式渲染，强调渲染。当您移动您的手时，该程序会以交互方式更新内部几何模型。但 Strachey 和 Douglas 的游戏也做到了这一点。重要的新事物是 Sketchpad在您移动您的手时将更新的模型呈现到显示器上。它不仅改变了模型。它也重新渲染了它。画板不知道下一条线可能在哪里绘制，不知道它会多长，也不知道它会处于什么角度。它必须像用户的手移动端点位置一样快地渲染它。因此，用户在交互式地更改几何模型方面具有完全的代理感。

This was really Sutherland’s innovation: Sketchpad rendered interactively, with emphasis on rendered. The program updated an internal geometric model interactively, as you moved your hand. But Strachey’s and Douglas’s games did that too. The important new thing was that Sketchpad rendered the updated model to the display while you moved your hand. It didn’t just change the model. It re-rendered it too. Sketchpad had no notion of where the next line might be drawn, nor how long it would be, nor at what angle it would lie. And it had to render it as fast as the user’s hand moved the endpoint locations. Thus the user had a sense of full agency in changing the geometric model interactively.

Strachey 的草稿程序在曼彻斯特古老的 Mark I 计算机上运行，​​显示了预渲染的几何元素——棋盘、Xs 和 Os。Mark I 的速度还不够快，甚至无法以交互速度渲染 X。道格拉斯在剑桥古老的 Edsac 上跑步的 noughts 和 crosss 也同样处于不利地位。但是大约10 年后，麻省理工学院的 TX-2速度足够快，可以瞬间渲染几十条线段，无论如何都是人类所感知的。

Strachey’s draughts program, running on the ancient Mark I computer at Manchester, displayed pre-rendered geometrical elements—the board, Xs, and Os. The Mark I wasn’t fast enough to render even an X at interactive speeds. Douglas’s noughts and crosses running on the ancient Edsac at Cambridge was similarly disadvantaged. But 10 or so years later TX-2 at MIT was fast enough to render several dozen line segments instantaneously, as perceived by a human anyway.

Sutherland 拥有巨大的 TX-2，并且是第一个以图形方式利用其速度的人。我们已经讨论过橡皮筋。他还让 Sketchpad 做一些事情，例如保持线段的一个端点固定，同时围绕它旋转另一个端点。效果是旋转线段——正如你所看到的——而用户（萨瑟兰）在显示器表面上移动光笔。或者他可以在旋转旋钮的同时实时旋转由多个线段组成的整个对象。这款游戏与 1950 年代初期的互动游戏的主要区别在于，线段的渲染是瞬间发生的，或者显然是瞬间发生的。Sketchpad 程序不知道新线段可能位于何处，因此无法预渲染它们。

Sutherland had giant TX-2 to himself and was the first to exploit its speed graphically. We’ve already discussed rubberbanding. He also caused Sketchpad to do things such as hold one endpoint of a line segment fixed while rotating the other endpoint about it. The effect was to rotate the line segment—as you watched—while the user (Sutherland) moved the light pen on the display’s surface. Or he could rotate an entire object, consisting of several line segments, in real time while twirling a knob. The key difference between this and interactive games in the early 1950s was that the rendering of line segments happened instantaneously, or apparently so. The Sketchpad program had no idea of where new line segments might be located, so couldn’t have pre-rendered them.

Sutherland 使用 Sketchpad 完成的是第一个交互式渲染的计算机图形。它不像 Sutherland 和 Sketchpad 经常被认为的“第一个交互式计算机图形”那样顺利地从舌头上滚下来，但它具有准确的优势。这个概念是渲染到显示器的几何模型立即被模型的另一个渲染替换，该模型包含依赖于用户输入的变化，因此程序无法预测。TX-2 是第一台具有足够快显示屏的单用户计算机，而 Sutherland 则跃跃欲试地采用了这种方式。

What Sutherland accomplished with Sketchpad was the first interactively rendered computer graphics. It doesn’t roll off the tongue as nicely as “first interactive computer graphics”—with which Sutherland and Sketchpad are often credited—but it has the advantage of being accurate. The notion is that a geometric model rendered to a display is instantaneously replaced with another rendering of the model, which incorporates changes that depend on a user’s input and hence can’t be predicted by the program. TX-2 was the first single-user computer with a display that was fast enough, and Sutherland leapt to use it that way.

但是 Sketchpad 有一些新的东西：交互式渲染使之成为可能但没有解释的东西；在这个交互式渲染的讨论中还没有捕捉到一些东西。好奇的？新的东西是这样的：用户自己感觉好像他（通常还不是她）正在通过“触摸”显示器上的图片来改变模型。我们称之为图片即界面的概念。早期的交互式棋盘游戏通过打字来交互地改变内部模型，而不是通过触摸图片本身。早期的 Edsac 游戏“羊与门”通过挥手中断 Edsac 纸带阅读器的光束来工作。但是作为Sketchpad的用户，新的体验是你自己通过触摸来改变图片. 如果您是设计师，您可以第一次在创建模型时看到它。DAC-1 几乎在同一时间允许同样的事情。

But there was something new about Sketchpad: something that interactive rendering made possible but doesn’t explain; something that hasn’t yet been captured in this discussion of interactive rendering. Curious? The something new was this: the user himself felt as if he (not yet she, generally) were changing the model by “touching” a picture of it on the display. We call this the picture-as-interface notion. The early interactive board games changed the internal model interactively by typing, not by touching the picture itself. And the early Edsac game called “sheep and gates” worked by interrupting the light beam of Edsac’s paper tape reader with a hand wave. But as a user of Sketchpad, the new experience was for you yourself to change the picture by touching it. If you were a designer you could, for the first time, see your model while it was being created. DAC-1 allowed the same thing at about the same time.

我在这里非常宽松地使用“触摸”，包括将光笔或光枪靠近计算机显示屏。由于使用鼠标进行光标操作对我们来说就像用光标触摸某物一样，所以我也将该模式包括为触摸。同样，我还包括通过在触摸板上移动轨迹球、操纵杆、手指或在虚拟现实中挥动魔杖来进行光标定位。

I’m using “touch” here quite loosely, to include bringing a light pen or light gun into close proximity to a computer display screen. Since cursoring with a mouse feels to us like touching something with the cursor, I also include that mode as touch. Similarly, I include cursoring by moving a trackball, joystick, your finger on a touchpad, or waving a wand in virtual reality.

实际上用一根或多根手指触摸屏幕，就像我们今天在手机和触摸板上所做的那样，似乎是过去十年左右的一个相当新的发展。但事实并非如此。由伊利诺伊大学的 Dan Alpert、Don Bitzer 和其他人策划的 Plato 项目从 1972 年开始让用户直接触摸其等离子面板显示器。这些二进制光栅显示器的显示元素以发出橙色的光而闻名。最近出版的这个项目的历史， Brian Dear的 The Friendly Orange Glow，应该会在现代计算历史中重建柏拉图的发展。72

Actually touching a screen with a finger or fingers, as we do on cellphones and touchpads today, would appear to be a fairly recent development of the last decade or so. But that’s not true. The Plato project masterminded by Dan Alpert, Don Bitzer, and others at the University of Illinois let users directly touch its plasma panel displays, starting in 1972. The display elements of these binary raster displays famously glowed orange. A recently published history of this project, The Friendly Orange Glow by Brian Dear, should reestablish the Plato development in modern computing history.72

在我向人们展示现场演示之前的早期，我会描述在桌面上移动鼠标是如何改变显示器的。我会在每次谈话中花大约 15 分钟来解释这确实有效。人们感到惊讶的是，如果你在一个地方移动并导致另一个地方的光标以严格的模仿方式移动，你会感觉好像你是在显示器上移动，而不是在桌面上移动。1980 年代初个人电脑的出现消除了这个问题。突然之间，每个人都从个人经验中知道了鼠标是如何连接到光标的。

In the early days before I could show people a live demo, I would describe how moving a mouse on a desktop made a change on the display. I would spend about 15 minutes in every talk explaining that this really worked. People were surprised that if you were moving in one place and causing a cursor in another place to move in rigid mimicry, it would feel as if you were moving up there on the display, not down on the desktop. The advent of personal computers in the early 1980s banished that question. Suddenly everyone just knew, from personal experience, how a mouse connected to a cursor.

无论您是通过移动鼠标还是通过实际触摸屏幕来使屏幕上发生某些事情，您都认为您的手是改变画面的代理。感觉是这样，但实际情况并非如此。实际发生的情况是：“触摸”设备导致向正在运行的程序发出命令——Sketchpad 说。该程序使计算机内存中的几何模型更改，然后这会导致更改后的模型重新显示（重新渲染）到刚刚“触摸”的显示。

Whether you cause something to happen on a screen by moving a mouse or by actually touching the screen, you perceive that your hand is the agent that changes the picture. It feels that way, but that’s not what actually happens. What actually happens is this: The “touching” device causes commands to be issued to a running program—Sketchpad say. The program causes the geometric model in computer memory to change, and then that causes a re-display of the changed model—a re-rendering of it—to the display that has just been “touched.”

从今天的角度来看，使用二维几何——例如正方形、三角形或圆形——听起来并不多，但萨瑟兰首先出现了，他必须解决无数问题。他必须创建一个数据结构来表示模型中对象的拓扑结构：这条线段连接到那个线段，并在运动过程中保持连接。

From today’s vantage point, working with two-dimensional geometry—such as a square, a triangle, or a circle—doesn’t sound like much, but Sutherland was there first and had to solve myriad problems. He had to create a data structure that represented the topology of the objects in his model: this line segment is connected to that one—and stays connected during movements.

当 TX-2 进行 Sketchpad 工作时，他必须让显示设备显示物体的图片。也就是说，他必须考虑一个与主计算进程分开运行的显示进程。这就是现代显示器的工作原理，但萨瑟兰不得不直面它。

He had to cause a display device to display a picture of the objects while the TX-2 was doing Sketchpad work. That is, he had to think about a display process running separately from the main computation process. This is how modern displays work, but Sutherland had to confront it head on.

他必须创建一种算法，将线段作为一组点呈现到显示器上。这与 1962 年 Bresenham 的著名算法做同样事情的时间大致相同，但 Sutherland 独立解决了它并且不考虑效率，因为 TX-2 非常快。

He had to create an algorithm for rendering a line segment to the display as a set of points. This was about the same time as Bresenham’s famous algorithm in 1962 for doing the same thing, but Sutherland solved it independently and without regard for efficiency because the TX-2 was so fast.

他必须弄清楚如何在显示器上跟踪光笔。他之后的人，比如蒂姆约翰逊，可以从许多已经解决的问题开始。仅二维几何听起来并没有多大贡献，但 Sutherland 还首次创建了交互式渲染计算机图形的管道（可能还记得 DAC-1）。

He had to figure out how to track a light pen on a display. People after him, like Tim Johnson, could just start with many of these problems already solved. Two-dimensional geometry alone doesn’t sound like much of a contribution, but Sutherland also created the plumbing of interactively rendered computer graphics for the first time (probably, remember DAC-1).

对于计算机辅助设计，他贡献了约束、拓扑保存、准确性和对象测试等重要概念，我们在此不再赘述。73

And for computer-aided design, he contributed the important notions of constraints, topology preservation, accuracy, and testing of his objects that we won’t further pursue here.73

但最重要的是，萨瑟兰向我们展示了通过“触摸”它的显示屏并看到某些东西瞬间发生变化的感觉来控制计算机的力量。萨瑟兰可能并不完全是第一个，但几乎没有人看到，甚至不知道更早的例子。例如，圣人是最高机密。DAC-1 是通用汽车内部的。但麻省理工学院在营销方面很聪明。画板的电影演示被广泛传播。您仍然可以在 YouTube 上观看 Alan Kay 讲述（事后）1962 年 Sketchpad 的电影演示，感受最初的兴奋。光是橡皮筋就足以震惊世界，在当时肯定是新事物。74

But most importantly, Sutherland showed us the power of feeling in control of a computer by “touching” its display and seeing something change instantaneously. Sutherland might not have entirely been first, but hardly anybody saw, or was even aware of, earlier instances. Sage, for example, was top secret. And DAC-1 was internal to GM. But MIT was smart about marketing. Film demos of Sketchpad were distributed far and wide. You can still watch Alan Kay on YouTube narrate (well after the fact) a 1962 film demo of Sketchpad and sense the original excitement. The rubberbanding alone was enough to astonish the world, and it was certainly new at the time.74

在 Sketchpad 中这种交互性的使用似乎是我们现在所说的图形用户界面的戏剧性起源。我在 Skype 通话中向 Sutherland 询问了这个问题。他还没准备好走那么远。他的程序允许用户通过光笔通过“触摸”来控制物体的图片。

This use of interactivity in Sketchpad appears to have been the dramatic origin of what we now call the graphical user interface. I asked Sutherland about this in our Skype call. He wasn’t ready to go that far. His program allowed a user to control the picture of an object through “touch” via a light pen.

给我们完整的图形用户界面的最后一个比喻是用户可以通过触摸显示器来控制任何东西，而不仅仅是几何图形的图片。对象。在用户触摸下发生变化的内部模型可以是，例如，用户计算机的文件或文件夹系统。文件夹的图标是计算机内存中实际文件夹的显示版本。它不是它的图片，因为那里没有任何东西可以用来制作图片。移动文件夹的图标实际上会将文件夹移动到内存中的某个新位置。

The final metaphor that gave us the full graphical user interface was the notion that a user could control anything through touching a display, not just pictures of geometrical objects. The internal model that changes under user touch could be, for example, the file or folder system of the user’s computer. An icon of a folder is the displayed version of an actual folder in the computer memory. It is not a picture of it, because there is nothing there to make a picture of. Moving the folder’s icon actually moves the folder somewhere new in the memory.

控制论机缘巧合：艺术家发现新生的电脑动画

Cybernetic Serendipity: Artists Discover Nascent Computer Animation

贝尔实验室是我们在 1963 年观看 Edward Zajac 的计算机动画电影的所在地。它也是早期计算机动画制作系统的所在地。1964 年，贝尔实验室的研究员 Ken Knowlton 设计了一种计算机语言，用于以光栅模式缓慢生成电影，一次一帧地拍摄。它后来被称为 Beflix。它不是交互式的。它假设一帧在“高分辨率”模式下被分成 252 x 184 个小方块。“在打字机模式下”的小字符可以写到每个方块上。相机是散焦的，因此角色是不同强度的斑点。也就是说，它们是粗略的散布像素，在网格上且没有重叠。一部由诺尔顿在 1964 年以这种方式制作的原始电影可以在网上看到。Zajac 和一名学生在 1964 年和 1965 年使用诺尔顿的语言制作的一个，也可以在网上看到。84

Bell Labs was home as we’ve seen to Edward Zajac’s computer animated film in 1963. It was also home to an early computer animation production system. In 1964 the Bell Labs researcher Ken Knowlton devised a computer language for slowly generating movies—in raster mode—one frame at a time to film. It came to be known as Beflix. It was not interactive. It assumed a frame was divided, in “high-resolution” mode, into 252 by 184 little squares. Tiny characters “in typewriter mode” could be written to each square. The camera was defocused so that the characters were blobs of different intensities. That is, they were crude spread pixels, on a grid and without overlap. A primitive movie made this way by Knowlton in 1964 can be seen online. One made by Zajac and a student in 1964 and 1965, using Knowlton’s language, can also be seen online.84

锣声在此响起。艺术家们开始与贝尔实验室和其他地方的技术人员合作。这些是受过艺术家训练的艺术家，而不是计算机科学家，他们嗅到了空气中的新事物。他们是一个新品种，他们将计算视为一种可以在雕塑、音乐、诗歌和绘画艺术中探索的新鲜物质。我们在下一章中再次见到的一位是斯坦·范德贝克，他是一位实验电影制作人，他在 1960 年代在贝尔实验室制作电影，与肯·诺尔顿合作。

And here a gong sounds. Artists began to work with the technologists at Bell Labs and other places. These were artists trained as artists, not computer scientists, who sniffed something new in the air. They were a new breed who perceived computation as a fresh substance to explore in sculpture, music, poetry, and pictorial art. One, whom we meet again in the next chapter, was Stan Vanderbeek, an experimental filmmaker, who made movies at Bell Labs in the 1960s, working with Ken Knowlton.

这是著名的六十年代，定义受到质疑。一些计算机科学家将自己视为艺术家，这在艺术界引起了沮丧。争论直到今天，我不会在这里解决它们。Ken Knowlton，无论你称他为艺术家与否，都开始了一生对各种展示元素的实验。他使用任何东西——多米诺骨牌、骰子、玩具车和贝壳——创作了数十张数码照片，以传播他的像素。1990 年代，在他的第一部电影之后 34 年对于散布像素，Knowlton 仍然在介质中工作——使用茶壶碎片作为显示元素（图 6.36）。85

This was the famous Sixties, and definitions were being questioned. Some computer scientists saw themselves as artists, causing dismay in the artistic community. Arguments rage to this day, and I won’t settle them here. Ken Knowlton, whether you call him an artist or not, began a lifetime of experimentation with various display elements. He created dozens of digital pictures using anything—dominoes, dice, toy cars, and seashells, say—to spread his pixels. In the 1990s, 34 years after his first movie with spread pixels, Knowlton was still working in the medium—with teapot shards as display elements (figure 6.36).85

尽管 1960 年代后期的许多人不认为计算机是一种艺术工具，但有一位女士认为，她 1968 年的展览已载入史册。Jasia Reichardt 的开创性展览 Cyber​​netic Serendipity于当年在伦敦开幕。图 6.37 中的目录封面（左）和海报（右）现在是收藏品。即便如此，她还是小心翼翼地避免称作品的创作者为“艺术家”。五十年后，这在某些圈子中仍然是一个雷区，实际诉讼悬而未决。86

Although many people in the late 1960s didn’t consider the computer an artistic tool, one woman did, and her 1968 exhibition has gone down in history. Jasia Reichardt’s seminal exhibition Cybernetic Serendipity opened in London that year. The catalog cover (left), and poster for it (right), in figure 6.37 are collector’s items now. Even then she carefully avoided calling the creators of the works “artists.” Fifty years later this is still a minefield in certain circles, with actual lawsuits pending.86

但是 Robin Forrest 没有问题地避开这个术语，尽管他在图 6.37 海报的右侧贡献了扭曲的线框补丁。“我从不认为我的形象是艺术，但她 [Reichardt] 做到了。所以，一定是艺术！但为什么是那个图像？” 87

But Robin Forrest has no problem eschewing the term, despite his having contributed the contorted wireframe patch on the right side of the figure 6.37 poster. “I never considered my image to be art, but she [Reichardt] did. So, it must be art! But why that image?”87

2005 年，Reichardt 采用了间接的方式来描述创作者，他重复了另一位作者的话：

In 2005 Reichardt took an indirect route to describing the creators, by repeating another author’s words:

他还指出，1967 年存在的大部分照片主要是出于业余爱好，他讨论了 Michael Noll、Charles Csuri、Jack Citron、Frieder Nake、乔治尼斯和惠普帕特森。作为计算机艺术史的先驱，我们今天对所有这些名字都很熟悉。. . . 计算机诗歌和艺术的可能性在 1949 年首次被提及。到 1950 年代初，它已成为大学和科学机构的话题，当计算机图形学问世时，艺术家已成为科学家、工程师、建筑师. 88

He also pointed out that most of the pictures in existence in 1967 were produced mainly as a hobby and he discussed the work of Michael Noll, Charles Csuri, Jack Citron, Frieder Nake, Georg Nees, and H. P. Paterson. All these names are familiar to us today as the pioneers of computer art history. . . . The possibility of computer poetry and art was first mentioned in 1949. By the beginning of the 1950s it was a topic of conversation at universities and scientific establishments, and by the time computer graphics arrived on the scene, the artists were scientists, engineers, architects.88

本章中的人物，除了罗宾福雷斯特（也作为《控制论机缘巧合》的创作者）之外，还有威廉·费特、肯·诺尔顿、斯坦·范德贝克和爱德华·扎亚克。89

People in this chapter, in addition to Robin Forrest, who also appeared as creators in Cybernetic Serendipity were William Fetter, Ken Knowlton, Stan Vanderbeek, and Edward Zajac.89

摩尔定律加速：放大成为超新星

The Moore’s Law Speedup: Amplification Goes Supernova

摩尔定律于 1965 年问世。它的宣布开启了计算加速的第二个时代。它带来了我们现在所知道的 Digital Light 成果。

Moore’s Law arrived in 1965. Its announcement ushered in Epoch 2 of computation speedup. It brought Digital Light as we now know it to fruition.

“法律”——因为它不是真正的法律——以世界上最成功的计算机芯片公司英特尔的联合创始人戈登·摩尔命名。英特尔是加利福尼亚硅谷的基石公司，其知识起源直接追溯到晶体管。晶体管导致了集成电路——单个芯片上的许多晶体管——以及摩尔定律。在摩尔定律之前，计算机变得越来越强大，因为它们变得越来越强大。之后，它们变小了。也就是说，它们变得更密集。

The “Law”—because it isn’t really a law—was named for Gordon Moore, cofounder of Intel, the most successful computer chip company in the world. Intel, a cornerstone company of California’s Silicon Valley, traces its intellectual origins directly to the transistor. The transistor led to the integrated circuit—lots of transistors on a single chip—and to Moore’s Law. Before Moore’s Law, computers got bigger as they got more powerful. After it, they got smaller. That is, they got denser.

摩尔这样制定他的“定律”：集成电路芯片上的元件数量每 18 个月翻一番。实际上，他先是说 12，然后是 24，但是平均卡住。毕竟这是“法律”，而不是法律。这只是 1965 年的一次观察——仅基于四个微薄的数据点。99

Moore formulated his “Law” this way: The number of components on an integrated circuit chip doubles every 18 months. Actually, he first said 12, then later 24, but the average stuck. It’s a “Law,” and not a law, after all. It was simply an observation in 1965—based on just four meager data points.99

他还预测——这只是事后看来很幽默——“没有理由相信它不会在至少 10 年内保持几乎不变。” 但它现在已经保持了55年。它已成为法律，尽管事实上没有理由以物理学为基础相信它确实是法律。100

He also predicted—it’s only humorous in hindsight—that “there is no reason to believe it will not remain nearly constant for at least 10 years.” But it’s held constant now for 55 years. It’s become the Law, despite the fact that there’s no reason grounded in physics to believe it really is one.100

我之前说过摩尔定律的一个方便的重新表述：关于计算机的所有优点每五年都会提高一个数量级：“10 Xin 5”很容易记住它（发音为“10 ex in 5”）。图 6.39 是 Epoch 2 的图片，左侧是 1965 年的第一年。

I stated a handy reformulation of Moore’s Law earlier: Everything good about computers gets an order of magnitude better every five years: “10X in 5” is an easy way to remember it (pronounced “ten ex in five”). Figure 6.39 is a picture of Epoch 2, with its Year One, 1965, on the left.

脚踏实地的商界称摩尔定律曲线为“曲棍球棒”。这是一个值得死去的利润图表。以显示的最后一个角度延伸曲线（接近垂直）以查看曲棍球棒。这是一个指数增长的画面。指数意味着爆炸。摩尔定律描述并预测了指数级增长计算速度或内存密度，或者成本和大小的指数级下降——计算机的所有优点都会以指数级的方式变得更好。

The down-to-earth business world calls the Moore’s Law curve a “hockey stick.” It’s a profits chart to die for. Extend the curve at the last angle shown—approaching vertical—to see the hockey stick. This is a picture of exponential increase. Exponential means explosive. Moore’s Law describes and predicts an exponential increase in computational speed or memory density, or an exponential decrease in cost and size—everything good about computers gets exponentially better.

摩尔定律是推动世界的阿基米德杠杆。人类能力的放大在第一纪元达到百万倍。然后摩尔定律在第二纪元中利用了另一个千亿倍——到 2025 年将达到万亿倍。自计算机发明以来，人类能力的放大将达到五亿分之一到那时折叠——18 个数量级！这是不可想象的 - 和愚蠢的 - 像天翻地覆一样，但它是真实的。

Moore’s Law is the Archimedean lever that has moved the world. Amplification of human capabilities reached a millionfold in Epoch 1. Then Moore’s Law in Epoch 2 has leveraged the factor another hundred-billionfold—a trillionfold by 2025. Amplification of human capabilities since the invention of the computer, all told, will have been a quintillionfold by then—18 orders of magnitude! That’s as unimaginable—and as silly—as skydillionfold, but it’s real.

任何现代计算方面的历史——尤其是数字光——都不能忽视摩尔定律的巨大作用。这是过去五年中所有特定发现和进步的爆炸性背景——没有它，其中大多数都是不可行的。随着我们在接下来的章节中继续阅读像素的传记，摩尔定律所扮演的角色将越来越明显。在这种情况下，概念的谱系并没有脱离历史时间——脱离摩尔定律时钟的沉重滴答声。

No history of any modern aspect of computing—of Digital Light in particular—can ignore the spectacular role of Moore’s Law. It’s the explosive background against which all specific discoveries and advances of the last five decades have taken place—most of them infeasible without it. As we proceed through the biography of the pixel in the next chapters, the role Moore’s Law plays will be increasingly evident. The genealogy of concepts in this case is not divorced from historical time—from the heavy ticking of the Moore’s Law clock.

我们也不能忽视当前的滴答作响。“定律”仍在发展超新星。事实上，它正在超越。一颗超新星的亮度可以比它的起源恒星亮 100 亿倍，但计算机很快就会比原来的机器强大一万亿倍——仅在 Epoch 2 中——。掌握摩尔定律的最高级概念，无论多么微不足道，都是理解计算和计算机以及数字光的核心。

Nor can we disregard that ticking clock in the present. The “Law” is still going supernova. In fact, it’s going beyond. A supernova can shine 10 billion times brighter than its originating star, but computers will soon be a trillion times more powerful—in Epoch 2 alone—than the original machines. Getting a handle, however slight, on the superlatives of Moore’s Law is at the heart of understanding computation and computers—and Digital Light.

我们已经看到了摩尔定律到底是什么——但为什么会这样呢？部分答案是，实现比特没有迫在眉睫的物理障碍。它只是某种东西的存在或不存在——比如电压。它没有大小。这只是一个区别，仅此而已。这是纯粹的信息。答案的其余部分必须反映人类可以创新的最大速度。给定技术的指数级改进——计算机芯片技术中的摩尔定律——衡量一大群有创造力的人在竞争下改进技术的最终速度，当没有最终的物理障碍阻碍它的改进时当技术必须以自己的方式付费时。

We’ve seen what Moore’s Law is exactly—but why is it? A partial answer is that there’s no imminent physical barrier to the realization of a bit. It’s merely the presence or absence of something—say a voltage. It has no size. It’s just a distinction and nothing more. It’s pure information. The rest of the answer must reflect the maximum rate at which humans can innovate. The exponential improvement of a given technology—Moore’s Law in the case of computer chip technology—measures the ultimate speed at which a large group of creative humans can proceed to improve a technology, under competition, when there is no ultimate physical barrier to its improvement and when the technology must pay its own way.

从表面上看，摩尔定律是一个谜。如果我们相信它，那么我们为什么要为干预步骤而烦恼呢？如果我们知道计算机将在三年内提高四倍，为什么还要先建立两倍的提高倍数呢？如果我们知道计算机将在 10 年内提高 100 倍，那么为什么还要为 5 年内只有 10 个版本的因素而烦恼呢？为什么不直接去更高的因素？

On the face of it, Moore’s Law is a puzzler. If we believe it, then why should we bother with the intervening steps? If we know that computers will improve by a factor of four in three years, why bother building the factor of two improvement first? If we know that computers will improve by a factor of 100 in 10 years, then why bother with the factor of only 10 version in 5 years? Why not go directly to the higher factor?

这个问题的极端形式使其荒谬性显而易见。如果我们在 1965 年就知道计算机在 2020 年会好上一千亿倍——就像现在一样——为什么我们是否费心制造在 1970 年好 10 倍、在 1975 年好 100 倍、在 1980 年好 1000 倍等等的计算机？为什么我们不直接开始制造 2020 年的机器，跳过所有年份和 pokey 机器之间的时间？这种形式的问题显示了缺陷。1965 年，假设我们对新的“法律”有信心，我们根本无法想象 2020 年的计算机——更不用说如何建造它了。工程师必须到达每个级别，然后才能开始想象到达下一个级别的方法，然后创建它。我们不能跳过中间步骤。

An extreme form of this question makes its absurdity obvious. If we knew in 1965 that computers would be a hundred billion times better in 2020—as they are—why did we bother to build the computers that were 10 times better in 1970, 100 times better in 1975, 1,000 times better in 1980, and so forth? Why didn’t we just proceed to build the 2020 machines, skipping all the years—and pokey machines—between? The question in this form shows the flaw. In 1965, assuming that we had faith in the new “Law,” we were simply unable to imagine the computer of 2020—much less how to build it. Engineers have to arrive at each level before they can even begin to imagine a way to reach the next one, and then create it. We can’t skip the intervening steps.

不，我们必须遵循摩尔定律逐步改进芯片技术的路径，才能想象下一个改进可能是什么。这就像生物的进化。微小的变化最终会聚合成巨大的变化——生物的生存能力在每一步都完好无损。从细菌到婴儿不可能一步一步直接飞跃。就像在达尔文进化论中一样，摩尔定律的变化提高了拟合度——在计算机的情况下朝着更便宜、更小、更密集和更快的方向发展。在任何一种情况下，我们都看不到累积的变化将走向何方，因为没有目的论。但是，当涉及到技术时，我们会实施每一步，看看它是否真的有效，然后获得勇气、洞察力和对工程的掌握，以进行下一步。这也是我们如何评估进行下一步所必需的金融赌博的方式。我们需要知道它在可行的范围内。

No, we have to follow the Moore’s Law path of incremental improvement of chip technology to even imagine what the next improvement could be. It’s like the evolution of living things. Tiny changes aggregate into eventually massive changes—with the viability of the living thing intact at every step. There couldn’t have been a direct one-step leap from a bacterium to a baby. Just as in Darwinian evolution, Moore’s Law changes improve the fit—toward cheaper, smaller, denser, and faster in the case of computers. And we can’t see where the accumulated changes are heading in either case because there is no teleology. But, when it comes to technology, we implement each step to see if it actually works, then gain the courage, the insight, and the engineering mastery to proceed to the next step. It’s also how we can assess the financial gamble that’s necessary to proceed to the next step. We need to know that it’s within feasible bounds.

摩尔定律将向我们表达的硬件现象描述为放大。曾经深不可测的变得可行，然后变得平凡。

Moore’s Law describes the hardware phenomenon that expresses itself to us as Amplification. What was once unfathomable becomes feasible, then ordinary.

再次萨瑟兰：虚拟现实的诞生

Sutherland Again: The Birth of Virtual Reality

带注释的爱丽丝梦游仙境，由马丁·加德纳于 1960 年撰写，是一代人的必备读物，他们着迷于自己的心灵旅行和神奇的蘑菇。所以毫不奇怪，Ivan Sutherland 对 1965 年终极展示的描述读起来像是一个迷幻的梦——并唤起了爱丽丝。1968 年，萨瑟兰在哈佛实现了这一梦想的第一次通过，这一举动巩固了他作为 60 年代计算机图形学主要领导者的名声。他称其为头戴式显示器. 佩戴它的用户会看到一个简单的 3D 计算机生成场景，即“虚拟现实”，与用户周围的真实世界相混合。虚拟现实 (VR) 是计算机生成部分的矛盾现代造词。VR 与现实世界的混合称为增强现实 (AR)，即由 VR “增加”或“改进”的现实。这两个术语都是对“现实”的奇怪看法，但一直存在。

The Annotated Alice in Wonderland, written in 1960 by Martin Gardner, was de rigueur reading for a generation fascinated with their own mind trips and magic mushrooms. So it’s no surprise that Ivan Sutherland’s description of the ultimate display in 1965 reads like a psychedelic dream—and invokes Alice. In an act that cemented his fame as the dominant leader of computer graphics in the Sixties, Sutherland implemented a first pass of that dream at Harvard in 1968. He called it a head-mounted display. A user who wore it was treated to a simple three-dimensional computer-generated scene, a “virtual reality,” mixed with the real world in the user’s surround. Virtual reality (VR) is the oxymoronic modern coinage for the computer-generated part. The mixture of a VR with the real world is called augmented reality (AR)—that is, reality “increased” or “improved” by the VR. Both these terms are odd takes on “reality” but have stuck.

有人认为 Sutherland 的头戴式显示器是 VR 或 AR 的首次实现。有早期的头戴式显示器，但这是第一个近乎实时地跟踪头部位置并调整虚拟场景以适应用户不断变化的视角的显示器。图 6.40 显示了它的光学系统（这个人不是Sutherland）。103

Some consider Sutherland’s head-mounted display the first realization of VR or AR. There had been earlier head-mounted displays, but this was the first that tracked head position and adjusted the virtual scene, in close to real time, to accommodate the user’s always changing perspective. Figure 6.40 shows its optics system (the man is not Sutherland).103

萨瑟兰的头戴式显示器远不能控制“物质的存在”，但它确实呈现了一个由直线段创建的三维世界的立体视图。这个视角是拉里·罗伯茨的视角，它结合了齐次坐标和矩阵操作。Sutherland 1963 年的 Sketchpad 没有采用计算机图形学的中心法则，但他 1968 年的头戴式显示器做到了。104

Sutherland’s head-mounted display was far short of controlling “the existence of matter,” but it did present a stereo view of a three-dimensional world created from straight line segments in perspective. And the perspective was Larry Roberts’s perspective, the one that combined homogeneous coordinates and matrix manipulations. Sutherland’s Sketchpad of 1963 didn’t embrace the Central Dogma of computer graphics, but his head-mounted display of 1968 did.104

摩尔定律使 VR 成为可能。头戴式显示器的发展早在 Epoch 2 之后——在 Gordon Moore 于 1965 年制定“法律”之后。事实上，Sutherland 在他的头戴式显示器设计中使用了集成电路芯片。105

Moore’s Law enabled VR. The head-mounted display development occurred early in Epoch 2—after Gordon Moore formulated the “Law” in 1965. Indeed, Sutherland used integrated circuit chips in his head-mounted display design.105

据传闻，萨瑟兰随后著名地声称，计算机图形学中的所有问题都已解决！这对于年复一年继续解决计算机图形问题的几代贡献者来说是一个惊喜，而且仍然如此——Siggraph 会议上到处都是他们的论文。在我与萨瑟兰的谈话中，他告诉我他永远不会发表那种声明。真正与他的说法背道而驰的是萨瑟兰在 1966 年写的一篇名为“计算机图形学中的十个未解决问题”的论文。但传闻中的说法与他给犹他州另一位杰出毕业生詹姆斯·卡吉亚的建议一致：

Sutherland then famously claimed, according to rumor, that all the problems in computer graphics had been solved! That’s a surprise to generations of contributors who have continued to solve computer graphics problems year after year, and still do—the Siggraph conference is full of their papers. During my conversation with Sutherland he told me he would never have made that statement. And truly belying the claim that he did is a 1966 paper Sutherland wrote called “Ten Unsolved Problems in Computer Graphics.” But the rumored claim is consistent with advice he gave James Kajiya, another outstanding Utah graduate:

进行研究的最好方法是选择一个每个人都认为非常困难的问题。找到一种全新的方式来看待它，让 90% 的事情变得简单。解决简单的 90%。然后其他人都会努力工作 10% 并参考你。106

The best way to do research is to pick a problem that everyone thinks is really hard. Find a fresh way of looking at it that makes 90% of it easy. Solve the easy 90%. Then everyone else will work on the hard 10% and reference you.106

萨瑟兰本人在一个完全不同的领域转向了一个非常困难的问题。他与 Carver Mead（创造了摩尔定律一词）合作，解决了设计越来越高密度的集成电路芯片的看似棘手的问题。也就是说，萨瑟兰将注意力转向保持摩尔定律的活力和良好。今天，他仍在俄勒冈州波特兰的办公室里解决问题的一个困难方面。107

Sutherland himself moved on to a very hard problem in a completely different sphere. He paired up with Carver Mead (who coined the term Moore’s Law) to solve the seemingly intractable problems of designing integrated circuit chips of higher and higher density. That is, Sutherland turned his attention to keeping Moore’s Law alive and well. He’s still working on a difficult aspect of the problem today from his office in Portland, Oregon.107

数码灯：从形状到阴影

Digital Light: From Shapes to Shades

我通过描述我在 1970 年代初向 Ravi Shankar 演示的演示来开始本章。我的目的是通过使用二维情况来推动基于几何的模型的概念，并通过在样条概念中反向显示采样定理来完善采样定理的重要性。最后，样条曲线作为点的插值或近似导致补丁作为插值或曲线的近似（实际上是样条曲线）。这引入了计算机图形学的三维世界。

I started this chapter by describing a demonstration I presented to Ravi Shankar in the early 1970s. My purpose was to drive home the notion of a model based on geometry by using a two-dimensional case and to round out the importance of the Sampling Theorem by showing it in reverse in the concept of spline. Finally, the spline as an interpolation or approximation of points led to the patch as an interpolation or approximation of curves (of splines, in fact). That introduced the three-dimensional world of computer graphics.

然后我回到旋风时代的末期，在这一章的剩余部分中，我们在纳粹和苏联的威胁以及美国对它们的大规模技术反应所催生的环境中度过了 1950 年代和 1960 年代。美国政府，尤其是 ARPA 和 NASA 提供了一个受保护和控制的世界，让计算机创新者茁壮成长。他们在受保护的环境中获得了慷慨的支持，没有通常的盈利和行业竞争负担。

I then returned to the end of the Whirlwind era and spent the rest of the chapter taking us through the 1950s and 1960s in a milieu spawned by the Nazi and Soviet threats and the massive American technological responses to them. The US government and especially ARPA and NASA provided the protected and controlled world that let the computer innovators thrive. They were given their heads in protected environments with generous support and without the usual burdens of profitability and industrial competition.

因为带有彩色笔触的 Ravi Shankar 演示来自 1970 年代，所以您可能会误以为本章的其余部分（大约 1950 年代和 1960 年代）有彩色图形。但它没有。我们只是现在，在 1960 年代末期，色彩即将开花，摩尔定律发电机已准备好驱动它。

Because the Ravi Shankar demonstration with its colored brushstrokes was from the 1970s, you might have been lulled into thinking that the rest of the chapter, about the 1950s and 1960s, had color graphics. But it didn’t. We are only now, at the end of the 1960s, at the point where color is about to flower, and the Moore’s Law dynamo is poised to drive it.

从本章中的形状和灰度图片开始，我们将在接下来的章节中学习颜色、阴影、透明度、纹理、光线、阴影等，一般归类为阴影图像。我们学习如何渲染单一颜色的阴影三角形区域，然后将其放大到完整的计算机图形荣耀。

From shapes and grayscale pictures in this chapter, we graduate in the next chapters to color, shades, transparency, texture, light, shadow, and so forth, generically classed as shaded images. We learn how to render a single colored, shaded triangle-shaped area and then Amplify it to full computer graphics glory.

我进一步将我们的注意力限制在计算机图形的计算机动画部分，我们最终将了解电影是如何从机器的内脏中产生的。

I further restrict our focus there to just the part of computer graphics that is computer animation, and we’ll finally learn how a movie springs from the guts of a machine.

摩尔定律 1X (1965–1970)

Moore’s Law 1X (1965–1970)